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 [1921]. 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 [3234] 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.
The remainder of this paper is organized as follows. Section 2 introduces the model of double-sided clipping DCO-OFDM and formulates the signal shaping optimization problem. Sections 3 and 4 study the bias and subcarrier power optimization, respectively. Section 5 evaluates the impact of upside clipping by establishing an inner connection. Section 6 provides the simulation results, and Section 7 concludes the whole study.

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 $\sum _{k=1}^{N-1}p_{k}/N=p_{s}$.

 

Fig. 1. Double-sided clipping DCO-OFDM system diagram.

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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:

$$s_{clip,n}=\textrm{clip}\left[s_{n}+b;A\right]=\begin{cases} A & s_{n}+b\ge A\\ s_{n}+b & 0<s_{n}+b<A\\ 0 & s_{n}+b\le 0, \end{cases}$$
where the linear dynamic range $A$ is given by $A\triangleq A_{sat}-A_{tov}$.

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

$$f_{s_{clip,n}}\left(w;b,p_{s}\right)=\frac{1}{\sqrt{p_{s}}}\phi\left(\frac{w-b}{\sqrt{p_{s}}}\right)\left(u\left(w\right)-u\left(w-A\right)\right)+Q\left(\frac{A-b}{\sqrt{p_{s}}}\right)\delta\left(w-A\right)+Q\left(\frac{b}{\sqrt{p_{s}}}\right)\delta\left(w\right),$$
where $\delta \left (\cdot \right )$ is the Dirac delta function, $u\left (\cdot \right )$ is the unit step function, and $\phi \left (\cdot \right )$ and $Q\left (\cdot \right )$ represent the PDF and complementary cumulative distribution function (CCDF) of a standard Gaussian distribution, respectively. At last, $s_{clip,n}$ is transformed by the digital-analog converter (DAC) as analog signals, which further drive LED to emit the optical signal.

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

$$Y_{k}=H_{k}S_{clip,k}+N_{k},$$
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

$$S_{clip,k}=aS_{k}+N_{clip,k}\quad 1\leq k\leq N-1,$$
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
$$\begin{aligned} p_{c}\left(b,p_{s}\right) & \triangleq p_{s}\left(Q\left(-\frac{b}{\sqrt{p_{s}}}\right)-Q\left(\frac{A-b}{\sqrt{p_{s}}}\right)\right)\left(Q\left(\frac{b}{\sqrt{p_{s}}}\right)+Q\left(\frac{A-b}{\sqrt{p_{s}}}\right)\right)\\ & -\left(\sqrt{p_{s}}\phi\left(\frac{b}{\sqrt{p_{s}}}\right)-\sqrt{p_{s}}\phi\left(\frac{A-b}{\sqrt{p_{s}}}\right)-bQ\left(\frac{b}{\sqrt{p_{s}}}\right)+\left(A-b\right)Q\left(\frac{A-b}{\sqrt{p_{s}}}\right)\right)^{2}\\ & +b^{2}Q\left(\frac{b}{\sqrt{p_{s}}}\right)+\left(A-b\right)^{2}Q\left(\frac{A-b}{\sqrt{p_{s}}}\right)-b\sqrt{p_{s}}\phi\left(\frac{b}{\sqrt{p_{s}}}\right)-\left(A-b\right)\sqrt{p_{s}}\phi\left(\frac{A-b}{\sqrt{p_{s}}}\right). \end{aligned}$$
Lemma 1 establishes a linear model to characterize the nonlinear double-sided clipping process via decomposing it into an attenuation term $a$ and a signal-uncorrelated noise $N_{clip,k}$. With the help of the clipping model Eq. (4), we can obtain a complete signal model in the frequency domain as
$$Y_{k}=aH_{k}S_{k}+H_{k}N_{clip,k}+N_{k},\quad 1\leq k\leq N-1.$$

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):

$$r\left(b,\left\{ p_{k}\right\} \right)\triangleq\sum_{k=1}^{N-1}\textrm{log}\left(1+\frac{p_{k}}{p_{s}}\textrm{SNDR}_{k}\left(b,p_{s}\right)\right),$$
where $\textrm {SNDR}_{k}\left (b,p_{s}\right )$ is the signal-to-noise-plus-distortion ratio at subcarrier $k$
$$\textrm{SNDR}_{k}\left(b,p_{s}\right)\triangleq\frac{|H_{k}|^{2}a^{2}\left(b,p_{s}\right)p_{s}}{|H_{k}|^{2}p_{c}\left(b,p_{s}\right)+p_{n}}.$$
Formally, the signal shaping optimization for double-sided clipping DCO-OFDM is formulated as
$$\begin{aligned} \mathbb{P}:\underset{b,\left\{ p_{k}\right\} }{\textrm{maximize}}\quad & r\left(b,\left\{ p_{k}\right\} \right)=\sum_{k=1}^{N-1}\textrm{log}\left(1+\frac{p_{k}}{p_{s}}\textrm{SNDR}_{k}\left(b,p_{s}\right)\right)\\ \textrm{subject to}\quad & \sum_{k=1}^{N-1}p_{k}/N=p_{s}.\end{aligned}$$
Remark that the signal shaping in $\mathbb {P}$ is never easy to solve because of its nonconvexity, nonlinearity, and complexity. Even though appearing similar to the single-sided clipping problem in [33], the double-sided clipping optimization in $\mathbb {P}$ is more challenging. Owing to the extra upside distortion, the expressions of the attenuation factor $a\left (b,p_{s}\right )$ and clipping noise power $p_{c}\left (b,p_{s}\right )$ become extremely complicated than those in the single-sided clipping problem. It explains why the shaping problem under the single-sided clipping has been well addressed, but the one under double-sided clipping remains open, especially for the optimal signal power $p_{s}$. In the following, we are going to take a deep dive into the signal shaping problem for double-sided clipping DCO-OFDM.

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

$$\begin{aligned}\frac{\partial}{\partial b}a\left(b,p_{s}\right) & \begin{cases} >0, & 0<b<A/2\\ <0, & A/2<b<A. \end{cases} \end{aligned}$$
The similar conclusion holds for $p_{c}\left (b,p_{s}\right )$. (Please see Fig. 2(a).) Therefore, $b^{\star }=A/2$ maximizes $a\left (b,p_{s}\right )$ and minimizes $p_{c}\left (b,p_{s}\right )$ simultaneously. According to the expression in Eq. (8), $b^{\star }=A/2$ is the unique point maximizing $\textrm {SNDR}_{k}\left (b,p_{s}\right )$ for each subcarrier. The maximization of $\textrm {SNDR}_{k}\left (b,p_{s}\right )$ for each subcarrier is equivalent to maximizing $r\left (b,\left \{ p_{k}\right \} \right )$ for any given $\left \{ p_{k}\right \}$. Therefore, $b^{\star }=A/2$ is also the unique maximum of $r\left (b,\left \{ p_{k}\right \} \right )$ within the dynamic range $\left [0,A\right ]$.

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.

 

Fig. 2. Numerical illustrations for bias optimization. (a) $a\left (b,p_{s}\right )$ and $p_{c}\left (b,p_{s}\right )$. (b) $r\left (b,\left \{ p_{k}\right \} \right )/\left (2N\right )$ and ${\cal I}\left (y_{n};s_{clip,n}|b,\left \{ p_{k}\right \} \right )$. (Configurations: $A=$700, $\sqrt {p_{k}}=$300, $\sqrt {p_{n}}=$30, $H_{k}=1$ and $2N=512$.)

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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

$${\cal I}\left(y_{n};s_{clip,n}|b,\left\{ p_{k}\right\} \right)={\cal I}\left(y_{n}^{\prime};s_{clip,n}^{\prime}|A-b,\left\{ p_{k}\right\} \right), $$
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

$$\begin{aligned} s_{clip,n} & \triangleq\textrm{clip}\left[s_{n}+b;A\right],\\ s_{clip,n}^{\prime} & \triangleq\textrm{clip}\left[A-b-s_{n};A\right]. \end{aligned}$$
These two signals $s_{clip,n}$ and $s_{clip,n}^{\prime }$ are exactly symmetric about $A/2$, since
$$\begin{aligned} s_{clip,n}^{\prime} & =\textrm{clip}\left[A-b-s_{n};A\right]\\ & =A-\textrm{clip}\left[s_{n}+b;A\right]=A-s_{clip,n}. \end{aligned}$$
Through linear VLC channels with the same impulse response, it is not difficult to show that the PDFs of the received signals $y_{n}$ and $y_{n}^{\prime }$ are also symmetric (about $AH_{0}/2$), where $H_{0}$ is the DC gain of the VLC channel. Thus, we can obtain that $y_{n}$ and $y_{n}^{\prime }$ have the same entropy, i.e., ${\cal H}\left (y_{n}\right )={\cal H}\left (y_{n}^{\prime }\right )$. Since the VLC channel has the same background noise level, we further have ${\cal H}\left (y_{n}|s_{clip,n}\right )={\cal H}\left (y_{n}^{\prime }|s_{clip,n}^{\prime }\right )$. (For example, in a flat channel case, both ${\cal H}\left (y_{n}|s_{clip,n}\right )$ and ${\cal H}\left (y_{n}^{\prime }|s_{clip,n}^{\prime }\right )$ depend on the distribution of Gaussian background noise only and thus equal to each other. The equality also holds for dispersive channels.) Therefore, we have the following result:
$$\begin{aligned} {\cal I}\left(y_{n};s_{clip,n}|b,\left\{ p_{k}\right\} \right) & ={\cal H}\left(y_{n}\right)-{\cal H}\left(y_{n}|s_{clip,n}\right)\\ & ={\cal H}\left(y_{n}^{\prime}\right)-{\cal H}\left(y_{n}^{\prime}|s_{clip,n}^{\prime}\right)={\cal I}\left(y_{n}^{\prime};s_{clip,n}^{\prime}|A-b,\left\{ p_{k}\right\} \right). \end{aligned}$$
That is, the mutual information ${\cal I}\left (y_{n};s_{clip,n}|b,\left \{ p_{k}\right \} \right )$ is symmetric about $b=A/2$.

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

$$ \frac{d}{dp_{s}}\textrm{SNDR}_{k}\left(A/2,p_{s}\right)=\frac{A\cdot\textrm{SNDR}_{k}^{2}\left(A/2,p_{s}\right)}{a^{3}p_{s}^{3/2}}q\left(p_{s}\right)f\left(p_{s};\gamma\right), $$
where $q\left (p_{s}\right )\triangleq \frac {A}{\sqrt {p_{s}}}\phi \left (\frac {A}{2\sqrt {p_{s}}}\right )+2Q\left (\frac {A}{2\sqrt {p_{s}}}\right )-1$. Since $\frac {d}{dp_{s}}q\left (p_{s}\right )=\frac {A^{3}}{8p_{s}^{5/2}}\phi \left (\frac {A}{2\sqrt {p_{s}}}\right )>0$, $q\left (p_{s}\right )$ is upper bounded by $\underset {p_{s}\rightarrow +\infty }{\lim }q\left (p_{s}\right )=2Q\left (0\right )-1=0$ and thus is negative. And obviously, $\frac {A\cdot \textrm {SNDR}_{k}^{2}\left (A/2,p_{s}\right )}{a^{3}p_{s}^{3/2}}$ is positive. Now let us look at the term $f\left (p_{s};\gamma \right )$. It is an increasing function with a unique positive root, because of
$$\begin{aligned} \underset{p_{s}\rightarrow 0^{+}}{\lim}f\left(p_{s};\gamma\right) & =-\frac{A}{4\gamma\sqrt{p_{s}}}<0,\underset{p_{s}\rightarrow+\infty}{\lim}f\left(p_{s};\gamma\right)=\phi\left(\frac{A}{2\sqrt{p_{s}}}\right)>0,\\ \textrm{and}, \frac{d}{dp_{s}}f\left(p_{s};\gamma\right) & =\frac{A}{4p_{s}^{3/2}}Q\left(\frac{A}{2\sqrt{p_{s}}}\right)+\frac{A}{8\gamma p_{s}^{3/2}}>0. \end{aligned}$$
Denote the unique root by $p_{s}^{\star }\left (\gamma \right )$. We have $\frac {d}{dp_{s}}\textrm {SNDR}_{k}\left (A/2,p_{s}\right )>0$ for $p_{s}<p_{s}^{\star }\left (\gamma \right )$ and $\frac {d}{dp_{s}}\textrm {SNDR}_{k}\left (A/2,p_{s}\right )<0$ for $p_{s}>p_{s}^{\star }\left (\gamma \right )$. Therefore, $\textrm {SNDR}_{k}\left (A/2,p_{s}\right )$ is maximized at $p_{s}^{\star }\left (\gamma \right )$, and so is $r\left (A/2,\left \{ p_{k}\right \} \right )$ for flat channels.

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

$$ r\left(b^{\star},p_{s},\left\{ u_{k}\right\} \right)\triangleq\sum_{k=1}^{N-1}\textrm{log}\left(1+u_{k}\textrm{SNDR}_{k}\left(b^{\star},p_{s}\right)\right). $$
Clearly, $u_{k}$ satisfies $\sum _{k=1}^{N-1}u_{k}=N$ according to the subcarrier power constraint $\sum _{k=1}^{N-1}p_{k}/N=p_{s}$.

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

$$\begin{aligned} r_{lb}\left(p_{s};b^{\star},p_{s}^{\prime},\left\{ u_{k}\right\} \right) & =\sum_{k=1}^{N-1}\textrm{log}\left(1+u_{k}\textrm{SNDR}_{k}\left(b^{\star},p_{s}^{\prime}\right)\right)\\ & -\sum_{k=1}^{N-1}\frac{u_{k}\textrm{SNDR}_{k}^{2}\left(b^{\star},p_{s}^{\prime}\right)}{1+u_{k}\textrm{SNDR}_{k}\left(b^{\star},p_{s}^{\prime}\right)}\left(\textrm{SNDR}_{k}^{-1}\left(b^{\star},p_{s}\right)-\textrm{SNDR}_{k}^{-1}\left(b^{\star},p_{s}^{\prime}\right)\right), \end{aligned}$$
where $p_{s}^{\prime }$ is a specific value of the signal power. We can guarantee that $r\left (b^{\star },p_{s},\left \{ u_{k}\right \} \right )\geq r_{lb}\left (p_{s};b^{\star },p_{s}^{\prime },\left \{ u_{k}\right \} \right )$ for any positive $p_{s}$ because of the first-order condition of a convex function:
$$ \log\left(1+1/y\right)\geq\log\left(1+1/x\right)-\frac{1}{x^{2}+x}\left(y-x\right). $$
Meanwhile, $r_{lb}\left (p_{s};b^{\star },p_{s}^{\prime },\left \{ u_{k}\right \} \right )$ is well designed so that it is tangent to $r\left (b^{\star },p_{s},\left \{ u_{k}\right \} \right )$ at the point $p_{s}^{\prime }$, i.e.,
$$\begin{aligned} r_{lb}\left(p_{s}^{\prime};b^{\star},p_{s}^{\prime},\left\{ u_{k}\right\} \right) & =r\left(b^{\star},p_{s}^{\prime},\left\{ u_{k}\right\} \right),\\ \frac{d}{dp_{s}}r_{lb}\left(p_{s};b^{\star},p_{s}^{\prime},\left\{ u_{k}\right\} \right) & \mid_{p_{s}=p_{s}^{\prime}}=\frac{\partial}{\partial p_{s}}r\left(b^{\star},p_{s},\left\{ u_{k}\right\} \right)\mid_{p_{s}=p_{s}^{\prime}}. \end{aligned}$$
Note that the above two properties hold for any positive $b$. However, the lower bound $r_{lb}\left (p_{s};b^{\star },p_{s}^{\prime },\left \{ u_{k}\right \} \right )$ is still challenging to be characterized. Fortunately, the results from Theorem 2 sheds light on optimizing $r_{lb}\left (p_{s};b^{\star },p_{s}^{\prime },\left \{ u_{k}\right \} \right )$, summarized in the following theorem.

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

$$ \frac{d}{dp_{s}}r_{lb}\left(p_{s};b^{\star},p_{s}^{\prime},\left\{ u_{k}\right\} \right)=\frac{Aq\left(p_{s}\right)}{a^{3}p_{s}^{3/2}}f_{d}\left(p_{s};\left\{ \gamma_{k}\right\} \right), $$
where $q\left (p_{s}\right )$ is negative and $f\left (p_{s};\gamma _{k}\right )$ is increasing in $p_{s}$ according to the proof of Theorem 2. Thus, $f_{d}\left (p_{s};\left \{ \gamma _{k}\right \} \right )$, the nonnegative weighted sum of $f\left (p_{s};\gamma _{k}\right )$, is also increasing in $p_{s}$. We can further prove that $f_{d}\left (p_{s};\left \{ \gamma _{k}\right \} \right )$ admits a unique positive root, denoted by $p_{s}^{*}\left (b^{\star },p_{s}^{\prime },\left \{ u_{k}\right \} \right )$, from $\underset {p_{s}\rightarrow 0^{+}}{\lim }f_{d}\left (p_{s};\left \{ \gamma _{k}\right \} \right )<0$ and $\underset {p_{s}\rightarrow +\infty }{\lim }f_{d}\left (p_{s};\left \{ \gamma _{k}\right \} \right )>0$. Therefore, $r_{lb}\left (p_{s};b^{\star },p_{s}^{\prime },\left \{ u_{k}\right \} \right )$ is a quasi-concave function with the unique maximizer $p_{s}^{*}\left (b^{\star },p_{s}^{\prime },\left \{ u_{k}\right \} \right )$.

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

$$u_{k}^{*}\left(p_{s}^{\prime}\right)=\left(\lambda\left(p_{s}^{\prime}\right)-\frac{1}{\textrm{SNDR}_{k}\left(b^{\star},p_{s}^{\prime}\right)}\right)^{+},$$
where $\left (\cdot \right )^{+}=\textrm {max}\left (\cdot ,0\right )$, and $\lambda \left (p_{s}^{\prime }\right )$ is the water level chosen to satisfy the power constraint.

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:

$$\begin{aligned} r\left(b^{\star},p_{s}^{i},\left\{ u_{k}^{i}\right\} \right)\overset{(a)}{=} & r_{lb}\left(p_{s}^{i};b^{\star},p_{s}^{i},\left\{ u_{k}^{i}\right\} \right)\overset{(b)}{\le}r_{lb}\left(p_{s}^{i+1};b^{\star},p_{s}^{i},\left\{ u_{k}^{i}\right\} \right)\\ \overset{(c)}{\le} & r\left(b^{\star},p_{s}^{i+1},\left\{ u_{k}^{i}\right\} \right)\overset{(d)}{\le}r\left(b^{\star},p_{s}^{i+1},\left\{ u_{k}^{i+1}\right\} \right), \end{aligned}$$
where (a) is because $r_{lb}\left (p_{s};b^{\star },p_{s}^{i},\left \{ u_{k}^{i}\right \} \right )$ is tangent to $r\left (b^{\star },p_{s},\left \{ u_{k}^{i}\right \} \right )$ at the point $p_{s}^{i}$, (b) follows from the maximality of $p_{s}^{i+1}$ in $r_{lb}\left (p_{s};b^{\star },p_{s}^{i},\left \{ u_{k}^{i}\right \} \right )$, (c) is because $r_{lb}\left (p_{s};b^{\star },p_{s}^{i},\left \{ u_{k}^{i}\right \} \right )$ serves a lower bound, and (d) is due to the optimality of waterfilling solution $\left \{ u_{k}^{i+1}\right \}$. We thus obtain a nondecreasing upper-bounded sequence, which is apparently convergent. Moreover, if the sequence $\left (b^{\star },p_{s}^{i},\left \{ u_{k}^{i}\right \} \right )$ generated by Algorithm 1 have limiting points, every limiting point is guaranteed to be a stationary point of the original problem [42]. We will demonstrate the performance and convergence properties of Algorithm 1 in Section 6.

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.

Tables Icon

Table 1. Parameters for simulations.

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

$$h^{(0)}(t;{\cal S},{\cal R})=\begin{cases} \frac{\mathcal{A}_{\textrm{PD}}\left(m+1\right)\cos^{m}\phi_{0}\cos\theta_{0}}{2\pi d_{0}^{2}}\delta\left(t-\frac{d_{0}}{c}\right), & 0\leq\theta_{0}\le\textrm{FOV}_{r}\\ 0, & \theta_{0}>\textrm{FOV}_{r}, \end{cases}$$
where $d_{0}$ is the distance between the source ${\cal S}$ and receiver ${\cal R}$, $c$ is the speed of light, $m$ is the Lambertian index related to the half-power semi-angle $\Phi _{1/2}$ of the LED given by $m=-\ln 2/\ln \left (\cos \left (\Phi _{1/2}\right )\right )$, $\mathcal {A}_{\textrm {PD}}$ is the detector size, $\textrm {FOV}_{r}$ is the receiver field-of-view (FOV) of the detector, and $\phi _{0}$ and $\theta _{0}$ are the angles of irradiance and incidence, respectively. Usually, the LOS channel can be approximated as a flat channel. We use the LOS impulse response $h^{(0)}(t;{\cal S},{\cal R})$ to represent the channel gain for the flat case.

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

$$ L_{1}=\frac{\mathcal{A}_{r}\left(m+1\right)\textrm{cos}^{m}\phi_{1}\textrm{cos}\theta_{1}}{2\pi d_{1}^{2}},\,L_{2}=\frac{\mathcal{A}_{r}\textrm{cos}\phi_{2}\textrm{cos}\theta_{2}}{\pi d_{2}^{2}},\,\ldots,\,L_{k+1}=\frac{\mathcal{A}_{\textrm{PD}}\textrm{cos}\phi_{k+1}\textrm{cos}\theta_{k+1}}{\pi d_{k+1}^{2}} $$
as the path-loss for the $k$th path, where $d_{k}$ is the distance between the source and the destination of the $k$th path, ${\cal A}_{r}$ is an elemental reflector, and $\phi _{k}$ and $\theta _{k}$ are the angles of irradiance and incidence for the $k$th path. According to [37,38], $h^{(k)}(t;{\cal S},{\cal R})$ can be expressed as
$$h^{(k)}(t;{\cal S},{\cal R})=\begin{cases} \int_{S}\left[L_{1}L_{2}\ldots L_{k+1}\Gamma^{\left(k\right)}\delta\left(t-\frac{d_{1}+d_{2}+\ldots+d_{k+1}}{c}\right)\right]d\mathcal{A}_{r}, & k\geq 1,0\leq\theta_{k+1}\leq\textrm{FOV}_{r}\\ 0, & \theta_{k+1}>\textrm{FOV}_{r}, \end{cases}$$
where $S$ is the surface of all reflectors, and $\Gamma ^{\left (k\right )}$ is the power of reflected light undergoing $k$ reflections from the LED. Assume that the reflectivity of different reflectors is constant and independent of the wavelength of the light. Thus, $\Gamma ^{\left (k\right )}$ is expressed as $\Gamma ^{\left (k\right )}=\rho _{1}\rho _{2}\ldots \rho _{k}$, where $\rho _{k}$ is the reflectivity of the corresponding elemental reflector for the $k$th reflection. Therefore, the channel impulse response (CIR) can be written as
$$h(t;{\cal S},{\cal R})=\sum_{k=0}^{\infty}h^{(k)}(t;{\cal S},{\cal R}).$$
The channel frequency response $H_{k}$ can be calculated via the FFT of the sampled impulse response $h(n\Delta t;{\cal S},{\cal R})$.

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

$$ g\left(t\right)=e^{-2\pi f_{b}t}, $$
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).
  • • Adaptive: the bias $b$ and signal power $p_{s}$ update in accord with specific OFDM symbols [24].
  • • 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.

 

Fig. 3. The optimal scheme and benchmarks over flat channels. (a) Achievable rate. (b) Signal shaping parameters.

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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.

 

Fig. 4. The optimal scheme and benchmarks over dispersive channels. (a) Achievable rate. (b) Convergence process of the optimal scheme ($\gamma =20$dB).

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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.

 

Fig. 5. Equivalent relationships between double-sided and single-sided clipping DCO-OFDM. (a) Optimal bias $b^{\star }$. (b) Optimal power $p_{s}^{\star }$.

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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.

 

Fig. 6. BER performance for different signal shaping schemes. (a) Uncoded DCO-OFDM. (b) Convolutional-coded DCO-OFDM. (c) Linear-coded DCO-OFDM.

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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.

 

Fig. 7. The constellation diagrams of different signal shaping schemes for uncoded DCO-OFDM at optical SNR $\gamma =25$dB. (a) Optimal. (b) Single-sided. (c) Approximated. (d) Adaptive. (e) $b=A/2,\sqrt {p_{s}}=A/8$. (f) $b=A/4,\sqrt {p_{s}}=A/8$.

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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.

References

1. H. Viswanathan and P. E. Mogensen, “Communications in the 6G era,” IEEE Access 8, 57063–57074 (2020). [CrossRef]  

2. M. Ayyash, H. Elgala, A. Khreishah, V. Jungnickel, T. Little, S. Shao, M. Rahaim, D. Schulz, J. Hilt, and R. Freund, “Coexistence of WiFi and LiFi toward 5G: Concepts, opportunities, and challenges,” IEEE Commun. Mag. 54(2), 64–71 (2016). [CrossRef]  

3. X. Li, R. Zhang, and L. Hanzo, “Optimization of visible-light optical wireless systems: Network-centric versus user-centric designs,” IEEE Commun. Surv. Tutorials 20(3), 1878–1904 (2018). [CrossRef]  

4. M. Z. Chowdhury, M. T. Hossan, A. Islam, and Y. M. Jang, “A comparative survey of optical wireless technologies: Architectures and applications,” IEEE Access 6, 9819–9840 (2018). [CrossRef]  

5. J. Armstrong, “OFDM for optical communications,” J. Lightwave Technol. 27(3), 189–204 (2009). [CrossRef]  

6. P. Ge, J. Wang, X. Ling, X. Liang, Y. Tian, and C. Zhao, “Achievable rate analysis for post- and pre-equalization in DCO-OFDM VLC with limited dynamic range,” Opt. Commun. (to be published).

7. S. D. Dissanayake and J. Armstrong, “Comparison of ACO-OFDM, DCO-OFDM and ADO-OFDM in IM/DD systems,” J. Lightwave Technol. 31(7), 1063–1072 (2013). [CrossRef]  

8. C.-H. Yeh, H.-Y. Chen, C.-W. Chow, and Y.-L. Liu, “Utilization of multi-band OFDM modulation to increase traffic rate of phosphor-LED wireless VLC,” Opt. Express 23(2), 1133–1138 (2015). [CrossRef]  

9. A. Adnan, Y. Liu, C.-W. Chow, and C.-H. Yeh, “Demonstration of non-Hermitian symmetry (NHS) serial-complex-valued orthogonal frequency division multiplexing (SCV-OFDM) for white-light visible light communication (VLC),” OSA Continuum 3(5), 1163–1168 (2020). [CrossRef]  

10. A. Adnan, Y. Liu, C. Chow, and C. Yeh, “Demonstration of non-Hermitian symmetry (NHS) IFFT/FFT size efficient OFDM non-orthogonal multiple access (NOMA) for visible light communication,” IEEE Photonics J. 12(3), 1–5 (2020). [CrossRef]  

11. M. S. Islim, R. X. Ferreira, X. He, E. Xie, S. Videv, S. Viola, S. Watson, N. Bamiedakis, R. V. Penty, I. H. White, A. E. Kelly, E. Gu, H. Haas, and M. D. Dawson, “Towards 10 Gb/s orthogonal frequency division multiplexing-based visible light communication using a GaN violet micro-LED,” Photonics Res. 5(2), A35–A43 (2017). [CrossRef]  

12. R. Bian, I. Tavakkolnia, and H. Haas, “15.73 Gb/s visible light communication with off-the-shelf LEDs,” J. Lightwave Technol. 37(10), 2418–2424 (2019). [CrossRef]  

13. D. Tsonev, S. Sinanovic, and H. Haas, “Complete modeling of nonlinear distortion in OFDM-based optical wireless communication,” J. Lightwave Technol. 31(18), 3064–3076 (2013). [CrossRef]  

14. S. Ma, R. Yang, Y. He, S. Lu, F. Zhou, N. Al-Dhahir, and S. Li, “Achieving channel capacity of visible light communication,” IEEE Systems Journal pp. 1–12 (2020).

15. J. Wang, Y. Xu, X. Ling, R. Zhang, Z. Ding, and C. Zhao, “PAPR analysis for OFDM visible light communication,” Opt. Express 24(24), 27457–27474 (2016). [CrossRef]  

16. Z. Lu, P. Liu, and S. Liu, “Experiment of beneficial clipping DCO-OFDM based visible light communication,” in Proceedings of IEEE 9th International Conference on Communication Software and Networks (IEEE, 2017), pp. 615–618.

17. J. Wu and Z. Wang, “Theoretical and experimental investigation of direct detection optical OFDM systems with clipping and normalization,” J. Electr. Comput. Eng. 2018, 1–11 (2018). [CrossRef]  

18. S. Dimitrov, S. Sinanovic, and H. Haas, “Clipping noise in OFDM-based optical wireless communication systems,” IEEE Trans. Commun. 60(4), 1072–1081 (2012). [CrossRef]  

19. S. Dimitrov, S. Sinanovic, and H. Haas, “Signal shaping and modulation for optical wireless communication,” J. Lightwave Technol. 30(9), 1319–1328 (2012). [CrossRef]  

20. S. Dimitrov and H. Haas, “Optimum signal shaping in OFDM-based optical wireless communication systems,” in Proceedings of IEEE Vehicular Technology Conference (IEEE, 2012), pp. 1–5.

21. S. Dimitrov and H. Haas, “Information rate of OFDM-based optical wireless communication systems with nonlinear distortion,” J. Lightwave Technol. 31(6), 918–929 (2013). [CrossRef]  

22. M. Zhang and Z. Zhang, “An optimum DC-biasing for DCO-OFDM system,” IEEE Commun. Lett. 18(8), 1351–1354 (2014). [CrossRef]  

23. M. Yu, Z. Zhang, L. Wu, and J. Dang, “Optimal symmetric double-sided signal clipping in DCO-OFDM visible light communication,” in Proceedings of Sixth International Conference on Wireless Communications and Signal Processing (2014), pp. 1–4.

24. Z. Wang, Q. Wang, S. Chen, and L. Hanzo, “An adaptive scaling and biasing scheme for OFDM-based visible light communication systems,” Opt. Express 22(10), 12707–12715 (2014). [CrossRef]  

25. R. Jiang, Q. Wang, F. Wang, L. Dai, and Z. Wang, “An optimal scaling scheme for DCO-OFDM based visible light communications,” Opt. Commun. 356, 136–140 (2015). [CrossRef]  

26. Y. He, M. Jiang, X. Ling, and C. Zhao, “Protograph-based EXIT analysis and optimization of LDPC coded DCO-OFDM in VLC systems,” IEEE Photonics Technol. Lett. 30(21), 1898–1901 (2018). [CrossRef]  

27. Y. He, M. Jiang, X. Ling, and C. Zhao, “A neural network aided approach for LDPC coded DCO-OFDM with clipping distortion,” in Proceedings of IEEE International Conference on Communications (IEEE, 2019), pp. 1–6.

28. Y. He, M. Jiang, X. Ling, and C. Zhao, “Robust BICM design for the LDPC coded DCO-OFDM: A deep learning approach,” IEEE Trans. Commun. 68(2), 713–727 (2020). [CrossRef]  

29. J. Cai, M. Chen, N. Huang, and J. Qian, “DC bias and power optimization for AV-DCO-OFDM in optical wireless communication,” Opt. Commun. (to be published).

30. W. Chu, J. Dang, Z. Zhang, and L. Wu, “Effect of clipping on the achievable rate of non-orthogonal multiple access with DCO-OFDM,” in Proceedings of 9th International Conference on Wireless Communications and Signal Processing (2017), pp. 1–6.

31. X. Huang, F. Yang, J. Song, and L. Cheng, “Subcarrier and power allocations for enhanced ADO-OFDM with dimming control,” in Proceedings of IEEE International Conference on Communications (IEEE, 2019), pp. 1–6.

32. X. Ling, J. Wang, X. Liang, Z. Ding, and C. Zhao, “Joint offset and power optimization for visible light DCO-OFDM systems,” in Proceedings of IEEE Global Communications Conference (IEEE, 2015), pp. 1–6.

33. X. Ling, J. Wang, X. Liang, Z. Ding, and C. Zhao, “Offset and power optimization for DCO-OFDM in visible light communication systems,” IEEE Trans. Signal Process. 64(2), 349–363 (2016). [CrossRef]  

34. L. Chen, B. Krongold, and J. Evans, “Theoretical characterization of nonlinear clipping effects in IM/DD optical OFDM systems,” IEEE Trans. Commun. 60(8), 2304–2312 (2012). [CrossRef]  

35. X. Ling, J. Wang, Z. Ding, C. Zhao, and X. Gao, “Efficient OFDMA for LiFi downlink,” J. Lightwave Technol. 36(10), 1928–1943 (2018). [CrossRef]  

36. X. Ling, J. Wang, X. Liang, Z. Ding, C. Zhao, and X. Gao, “Biased multi-LED beamforming for multicarrier visible light communications,” IEEE J. Sel. Areas Commun. 36(1), 106–120 (2018). [CrossRef]  

37. J. R. Barry, J. M. Kahn, W. J. Krause, E. A. Lee, and D. G. Messerschmitt, “Simulation of multipath impulse response for indoor wireless optical channels,” IEEE J. Sel. Areas Commun. 11(3), 367–379 (1993). [CrossRef]  

38. K. Lee, H. Park, and J. R. Barry, “Indoor channel characteristics for visible light communications,” IEEE Commun. Lett. 15(2), 217–219 (2011). [CrossRef]  

39. D. Dardari, V. Tralli, and A. Vaccari, “A theoretical characterization of nonlinear distortion effects in OFDM systems,” IEEE Trans. Commun. 48(10), 1755–1764 (2000). [CrossRef]  

40. A. V. Oppenheim, Discrete-Time Signal Processing (Pearson Education India, 1999).

41. J. Dennis and R. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (SIAM, 1996).

42. M. Razaviyayn, M. Hong, and Z.-Q. Luo, “A unified convergence analysis of block successive minimization methods for nonsmooth optimization,” SIAM J. Optim. 23(2), 1126–1153 (2013). [CrossRef]  

43. OSRAM GmbH, “Datasheet: LE UW S2LN,” (2011).

44. L. Wu, Z. Zhang, J. Dang, and H. Liu, “Adaptive modulation schemes for visible light communications,” J. Lightwave Technol. 33(1), 117–125 (2015). [CrossRef]  

45. D. Karunatilaka, F. Zafar, V. Kalavally, and R. Parthiban, “LED based indoor visible light communications: State of the art,” IEEE Commun. Surv. Tutorials 17(3), 1649–1678 (2015). [CrossRef]  

46. LED Engin, “Datasheet: LZ4-00MA00 RGBA LED,” (2016).

References

  • View by:
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  • |
  • |

  1. H. Viswanathan and P. E. Mogensen, “Communications in the 6G era,” IEEE Access 8, 57063–57074 (2020).
    [Crossref]
  2. M. Ayyash, H. Elgala, A. Khreishah, V. Jungnickel, T. Little, S. Shao, M. Rahaim, D. Schulz, J. Hilt, and R. Freund, “Coexistence of WiFi and LiFi toward 5G: Concepts, opportunities, and challenges,” IEEE Commun. Mag. 54(2), 64–71 (2016).
    [Crossref]
  3. X. Li, R. Zhang, and L. Hanzo, “Optimization of visible-light optical wireless systems: Network-centric versus user-centric designs,” IEEE Commun. Surv. Tutorials 20(3), 1878–1904 (2018).
    [Crossref]
  4. M. Z. Chowdhury, M. T. Hossan, A. Islam, and Y. M. Jang, “A comparative survey of optical wireless technologies: Architectures and applications,” IEEE Access 6, 9819–9840 (2018).
    [Crossref]
  5. J. Armstrong, “OFDM for optical communications,” J. Lightwave Technol. 27(3), 189–204 (2009).
    [Crossref]
  6. P. Ge, J. Wang, X. Ling, X. Liang, Y. Tian, and C. Zhao, “Achievable rate analysis for post- and pre-equalization in DCO-OFDM VLC with limited dynamic range,” Opt. Commun. (to be published).
  7. S. D. Dissanayake and J. Armstrong, “Comparison of ACO-OFDM, DCO-OFDM and ADO-OFDM in IM/DD systems,” J. Lightwave Technol. 31(7), 1063–1072 (2013).
    [Crossref]
  8. C.-H. Yeh, H.-Y. Chen, C.-W. Chow, and Y.-L. Liu, “Utilization of multi-band OFDM modulation to increase traffic rate of phosphor-LED wireless VLC,” Opt. Express 23(2), 1133–1138 (2015).
    [Crossref]
  9. A. Adnan, Y. Liu, C.-W. Chow, and C.-H. Yeh, “Demonstration of non-Hermitian symmetry (NHS) serial-complex-valued orthogonal frequency division multiplexing (SCV-OFDM) for white-light visible light communication (VLC),” OSA Continuum 3(5), 1163–1168 (2020).
    [Crossref]
  10. A. Adnan, Y. Liu, C. Chow, and C. Yeh, “Demonstration of non-Hermitian symmetry (NHS) IFFT/FFT size efficient OFDM non-orthogonal multiple access (NOMA) for visible light communication,” IEEE Photonics J. 12(3), 1–5 (2020).
    [Crossref]
  11. M. S. Islim, R. X. Ferreira, X. He, E. Xie, S. Videv, S. Viola, S. Watson, N. Bamiedakis, R. V. Penty, I. H. White, A. E. Kelly, E. Gu, H. Haas, and M. D. Dawson, “Towards 10 Gb/s orthogonal frequency division multiplexing-based visible light communication using a GaN violet micro-LED,” Photonics Res. 5(2), A35–A43 (2017).
    [Crossref]
  12. R. Bian, I. Tavakkolnia, and H. Haas, “15.73 Gb/s visible light communication with off-the-shelf LEDs,” J. Lightwave Technol. 37(10), 2418–2424 (2019).
    [Crossref]
  13. D. Tsonev, S. Sinanovic, and H. Haas, “Complete modeling of nonlinear distortion in OFDM-based optical wireless communication,” J. Lightwave Technol. 31(18), 3064–3076 (2013).
    [Crossref]
  14. S. Ma, R. Yang, Y. He, S. Lu, F. Zhou, N. Al-Dhahir, and S. Li, “Achieving channel capacity of visible light communication,” IEEE Systems Journal pp. 1–12 (2020).
  15. J. Wang, Y. Xu, X. Ling, R. Zhang, Z. Ding, and C. Zhao, “PAPR analysis for OFDM visible light communication,” Opt. Express 24(24), 27457–27474 (2016).
    [Crossref]
  16. Z. Lu, P. Liu, and S. Liu, “Experiment of beneficial clipping DCO-OFDM based visible light communication,” in Proceedings of IEEE 9th International Conference on Communication Software and Networks (IEEE, 2017), pp. 615–618.
  17. J. Wu and Z. Wang, “Theoretical and experimental investigation of direct detection optical OFDM systems with clipping and normalization,” J. Electr. Comput. Eng. 2018, 1–11 (2018).
    [Crossref]
  18. S. Dimitrov, S. Sinanovic, and H. Haas, “Clipping noise in OFDM-based optical wireless communication systems,” IEEE Trans. Commun. 60(4), 1072–1081 (2012).
    [Crossref]
  19. S. Dimitrov, S. Sinanovic, and H. Haas, “Signal shaping and modulation for optical wireless communication,” J. Lightwave Technol. 30(9), 1319–1328 (2012).
    [Crossref]
  20. S. Dimitrov and H. Haas, “Optimum signal shaping in OFDM-based optical wireless communication systems,” in Proceedings of IEEE Vehicular Technology Conference (IEEE, 2012), pp. 1–5.
  21. S. Dimitrov and H. Haas, “Information rate of OFDM-based optical wireless communication systems with nonlinear distortion,” J. Lightwave Technol. 31(6), 918–929 (2013).
    [Crossref]
  22. M. Zhang and Z. Zhang, “An optimum DC-biasing for DCO-OFDM system,” IEEE Commun. Lett. 18(8), 1351–1354 (2014).
    [Crossref]
  23. M. Yu, Z. Zhang, L. Wu, and J. Dang, “Optimal symmetric double-sided signal clipping in DCO-OFDM visible light communication,” in Proceedings of Sixth International Conference on Wireless Communications and Signal Processing (2014), pp. 1–4.
  24. Z. Wang, Q. Wang, S. Chen, and L. Hanzo, “An adaptive scaling and biasing scheme for OFDM-based visible light communication systems,” Opt. Express 22(10), 12707–12715 (2014).
    [Crossref]
  25. R. Jiang, Q. Wang, F. Wang, L. Dai, and Z. Wang, “An optimal scaling scheme for DCO-OFDM based visible light communications,” Opt. Commun. 356, 136–140 (2015).
    [Crossref]
  26. Y. He, M. Jiang, X. Ling, and C. Zhao, “Protograph-based EXIT analysis and optimization of LDPC coded DCO-OFDM in VLC systems,” IEEE Photonics Technol. Lett. 30(21), 1898–1901 (2018).
    [Crossref]
  27. Y. He, M. Jiang, X. Ling, and C. Zhao, “A neural network aided approach for LDPC coded DCO-OFDM with clipping distortion,” in Proceedings of IEEE International Conference on Communications (IEEE, 2019), pp. 1–6.
  28. Y. He, M. Jiang, X. Ling, and C. Zhao, “Robust BICM design for the LDPC coded DCO-OFDM: A deep learning approach,” IEEE Trans. Commun. 68(2), 713–727 (2020).
    [Crossref]
  29. J. Cai, M. Chen, N. Huang, and J. Qian, “DC bias and power optimization for AV-DCO-OFDM in optical wireless communication,” Opt. Commun. (to be published).
  30. W. Chu, J. Dang, Z. Zhang, and L. Wu, “Effect of clipping on the achievable rate of non-orthogonal multiple access with DCO-OFDM,” in Proceedings of 9th International Conference on Wireless Communications and Signal Processing (2017), pp. 1–6.
  31. X. Huang, F. Yang, J. Song, and L. Cheng, “Subcarrier and power allocations for enhanced ADO-OFDM with dimming control,” in Proceedings of IEEE International Conference on Communications (IEEE, 2019), pp. 1–6.
  32. X. Ling, J. Wang, X. Liang, Z. Ding, and C. Zhao, “Joint offset and power optimization for visible light DCO-OFDM systems,” in Proceedings of IEEE Global Communications Conference (IEEE, 2015), pp. 1–6.
  33. X. Ling, J. Wang, X. Liang, Z. Ding, and C. Zhao, “Offset and power optimization for DCO-OFDM in visible light communication systems,” IEEE Trans. Signal Process. 64(2), 349–363 (2016).
    [Crossref]
  34. L. Chen, B. Krongold, and J. Evans, “Theoretical characterization of nonlinear clipping effects in IM/DD optical OFDM systems,” IEEE Trans. Commun. 60(8), 2304–2312 (2012).
    [Crossref]
  35. X. Ling, J. Wang, Z. Ding, C. Zhao, and X. Gao, “Efficient OFDMA for LiFi downlink,” J. Lightwave Technol. 36(10), 1928–1943 (2018).
    [Crossref]
  36. X. Ling, J. Wang, X. Liang, Z. Ding, C. Zhao, and X. Gao, “Biased multi-LED beamforming for multicarrier visible light communications,” IEEE J. Sel. Areas Commun. 36(1), 106–120 (2018).
    [Crossref]
  37. J. R. Barry, J. M. Kahn, W. J. Krause, E. A. Lee, and D. G. Messerschmitt, “Simulation of multipath impulse response for indoor wireless optical channels,” IEEE J. Sel. Areas Commun. 11(3), 367–379 (1993).
    [Crossref]
  38. K. Lee, H. Park, and J. R. Barry, “Indoor channel characteristics for visible light communications,” IEEE Commun. Lett. 15(2), 217–219 (2011).
    [Crossref]
  39. D. Dardari, V. Tralli, and A. Vaccari, “A theoretical characterization of nonlinear distortion effects in OFDM systems,” IEEE Trans. Commun. 48(10), 1755–1764 (2000).
    [Crossref]
  40. A. V. Oppenheim, Discrete-Time Signal Processing (Pearson Education India, 1999).
  41. J. Dennis and R. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (SIAM, 1996).
  42. M. Razaviyayn, M. Hong, and Z.-Q. Luo, “A unified convergence analysis of block successive minimization methods for nonsmooth optimization,” SIAM J. Optim. 23(2), 1126–1153 (2013).
    [Crossref]
  43. OSRAM GmbH, “Datasheet: LE UW S2LN,” (2011).
  44. L. Wu, Z. Zhang, J. Dang, and H. Liu, “Adaptive modulation schemes for visible light communications,” J. Lightwave Technol. 33(1), 117–125 (2015).
    [Crossref]
  45. D. Karunatilaka, F. Zafar, V. Kalavally, and R. Parthiban, “LED based indoor visible light communications: State of the art,” IEEE Commun. Surv. Tutorials 17(3), 1649–1678 (2015).
    [Crossref]
  46. LED Engin, “Datasheet: LZ4-00MA00 RGBA LED,” (2016).

2020 (4)

H. Viswanathan and P. E. Mogensen, “Communications in the 6G era,” IEEE Access 8, 57063–57074 (2020).
[Crossref]

A. Adnan, Y. Liu, C.-W. Chow, and C.-H. Yeh, “Demonstration of non-Hermitian symmetry (NHS) serial-complex-valued orthogonal frequency division multiplexing (SCV-OFDM) for white-light visible light communication (VLC),” OSA Continuum 3(5), 1163–1168 (2020).
[Crossref]

A. Adnan, Y. Liu, C. Chow, and C. Yeh, “Demonstration of non-Hermitian symmetry (NHS) IFFT/FFT size efficient OFDM non-orthogonal multiple access (NOMA) for visible light communication,” IEEE Photonics J. 12(3), 1–5 (2020).
[Crossref]

Y. He, M. Jiang, X. Ling, and C. Zhao, “Robust BICM design for the LDPC coded DCO-OFDM: A deep learning approach,” IEEE Trans. Commun. 68(2), 713–727 (2020).
[Crossref]

2019 (1)

2018 (6)

J. Wu and Z. Wang, “Theoretical and experimental investigation of direct detection optical OFDM systems with clipping and normalization,” J. Electr. Comput. Eng. 2018, 1–11 (2018).
[Crossref]

X. Li, R. Zhang, and L. Hanzo, “Optimization of visible-light optical wireless systems: Network-centric versus user-centric designs,” IEEE Commun. Surv. Tutorials 20(3), 1878–1904 (2018).
[Crossref]

M. Z. Chowdhury, M. T. Hossan, A. Islam, and Y. M. Jang, “A comparative survey of optical wireless technologies: Architectures and applications,” IEEE Access 6, 9819–9840 (2018).
[Crossref]

Y. He, M. Jiang, X. Ling, and C. Zhao, “Protograph-based EXIT analysis and optimization of LDPC coded DCO-OFDM in VLC systems,” IEEE Photonics Technol. Lett. 30(21), 1898–1901 (2018).
[Crossref]

X. Ling, J. Wang, Z. Ding, C. Zhao, and X. Gao, “Efficient OFDMA for LiFi downlink,” J. Lightwave Technol. 36(10), 1928–1943 (2018).
[Crossref]

X. Ling, J. Wang, X. Liang, Z. Ding, C. Zhao, and X. Gao, “Biased multi-LED beamforming for multicarrier visible light communications,” IEEE J. Sel. Areas Commun. 36(1), 106–120 (2018).
[Crossref]

2017 (1)

M. S. Islim, R. X. Ferreira, X. He, E. Xie, S. Videv, S. Viola, S. Watson, N. Bamiedakis, R. V. Penty, I. H. White, A. E. Kelly, E. Gu, H. Haas, and M. D. Dawson, “Towards 10 Gb/s orthogonal frequency division multiplexing-based visible light communication using a GaN violet micro-LED,” Photonics Res. 5(2), A35–A43 (2017).
[Crossref]

2016 (3)

M. Ayyash, H. Elgala, A. Khreishah, V. Jungnickel, T. Little, S. Shao, M. Rahaim, D. Schulz, J. Hilt, and R. Freund, “Coexistence of WiFi and LiFi toward 5G: Concepts, opportunities, and challenges,” IEEE Commun. Mag. 54(2), 64–71 (2016).
[Crossref]

J. Wang, Y. Xu, X. Ling, R. Zhang, Z. Ding, and C. Zhao, “PAPR analysis for OFDM visible light communication,” Opt. Express 24(24), 27457–27474 (2016).
[Crossref]

X. Ling, J. Wang, X. Liang, Z. Ding, and C. Zhao, “Offset and power optimization for DCO-OFDM in visible light communication systems,” IEEE Trans. Signal Process. 64(2), 349–363 (2016).
[Crossref]

2015 (4)

R. Jiang, Q. Wang, F. Wang, L. Dai, and Z. Wang, “An optimal scaling scheme for DCO-OFDM based visible light communications,” Opt. Commun. 356, 136–140 (2015).
[Crossref]

L. Wu, Z. Zhang, J. Dang, and H. Liu, “Adaptive modulation schemes for visible light communications,” J. Lightwave Technol. 33(1), 117–125 (2015).
[Crossref]

D. Karunatilaka, F. Zafar, V. Kalavally, and R. Parthiban, “LED based indoor visible light communications: State of the art,” IEEE Commun. Surv. Tutorials 17(3), 1649–1678 (2015).
[Crossref]

C.-H. Yeh, H.-Y. Chen, C.-W. Chow, and Y.-L. Liu, “Utilization of multi-band OFDM modulation to increase traffic rate of phosphor-LED wireless VLC,” Opt. Express 23(2), 1133–1138 (2015).
[Crossref]

2014 (2)

2013 (4)

2012 (3)

L. Chen, B. Krongold, and J. Evans, “Theoretical characterization of nonlinear clipping effects in IM/DD optical OFDM systems,” IEEE Trans. Commun. 60(8), 2304–2312 (2012).
[Crossref]

S. Dimitrov, S. Sinanovic, and H. Haas, “Clipping noise in OFDM-based optical wireless communication systems,” IEEE Trans. Commun. 60(4), 1072–1081 (2012).
[Crossref]

S. Dimitrov, S. Sinanovic, and H. Haas, “Signal shaping and modulation for optical wireless communication,” J. Lightwave Technol. 30(9), 1319–1328 (2012).
[Crossref]

2011 (1)

K. Lee, H. Park, and J. R. Barry, “Indoor channel characteristics for visible light communications,” IEEE Commun. Lett. 15(2), 217–219 (2011).
[Crossref]

2009 (1)

2000 (1)

D. Dardari, V. Tralli, and A. Vaccari, “A theoretical characterization of nonlinear distortion effects in OFDM systems,” IEEE Trans. Commun. 48(10), 1755–1764 (2000).
[Crossref]

1993 (1)

J. R. Barry, J. M. Kahn, W. J. Krause, E. A. Lee, and D. G. Messerschmitt, “Simulation of multipath impulse response for indoor wireless optical channels,” IEEE J. Sel. Areas Commun. 11(3), 367–379 (1993).
[Crossref]

Adnan, A.

A. Adnan, Y. Liu, C.-W. Chow, and C.-H. Yeh, “Demonstration of non-Hermitian symmetry (NHS) serial-complex-valued orthogonal frequency division multiplexing (SCV-OFDM) for white-light visible light communication (VLC),” OSA Continuum 3(5), 1163–1168 (2020).
[Crossref]

A. Adnan, Y. Liu, C. Chow, and C. Yeh, “Demonstration of non-Hermitian symmetry (NHS) IFFT/FFT size efficient OFDM non-orthogonal multiple access (NOMA) for visible light communication,” IEEE Photonics J. 12(3), 1–5 (2020).
[Crossref]

Al-Dhahir, N.

S. Ma, R. Yang, Y. He, S. Lu, F. Zhou, N. Al-Dhahir, and S. Li, “Achieving channel capacity of visible light communication,” IEEE Systems Journal pp. 1–12 (2020).

Armstrong, J.

Ayyash, M.

M. Ayyash, H. Elgala, A. Khreishah, V. Jungnickel, T. Little, S. Shao, M. Rahaim, D. Schulz, J. Hilt, and R. Freund, “Coexistence of WiFi and LiFi toward 5G: Concepts, opportunities, and challenges,” IEEE Commun. Mag. 54(2), 64–71 (2016).
[Crossref]

Bamiedakis, N.

M. S. Islim, R. X. Ferreira, X. He, E. Xie, S. Videv, S. Viola, S. Watson, N. Bamiedakis, R. V. Penty, I. H. White, A. E. Kelly, E. Gu, H. Haas, and M. D. Dawson, “Towards 10 Gb/s orthogonal frequency division multiplexing-based visible light communication using a GaN violet micro-LED,” Photonics Res. 5(2), A35–A43 (2017).
[Crossref]

Barry, J. R.

K. Lee, H. Park, and J. R. Barry, “Indoor channel characteristics for visible light communications,” IEEE Commun. Lett. 15(2), 217–219 (2011).
[Crossref]

J. R. Barry, J. M. Kahn, W. J. Krause, E. A. Lee, and D. G. Messerschmitt, “Simulation of multipath impulse response for indoor wireless optical channels,” IEEE J. Sel. Areas Commun. 11(3), 367–379 (1993).
[Crossref]

Bian, R.

Cai, J.

J. Cai, M. Chen, N. Huang, and J. Qian, “DC bias and power optimization for AV-DCO-OFDM in optical wireless communication,” Opt. Commun. (to be published).

Chen, H.-Y.

Chen, L.

L. Chen, B. Krongold, and J. Evans, “Theoretical characterization of nonlinear clipping effects in IM/DD optical OFDM systems,” IEEE Trans. Commun. 60(8), 2304–2312 (2012).
[Crossref]

Chen, M.

J. Cai, M. Chen, N. Huang, and J. Qian, “DC bias and power optimization for AV-DCO-OFDM in optical wireless communication,” Opt. Commun. (to be published).

Chen, S.

Cheng, L.

X. Huang, F. Yang, J. Song, and L. Cheng, “Subcarrier and power allocations for enhanced ADO-OFDM with dimming control,” in Proceedings of IEEE International Conference on Communications (IEEE, 2019), pp. 1–6.

Chow, C.

A. Adnan, Y. Liu, C. Chow, and C. Yeh, “Demonstration of non-Hermitian symmetry (NHS) IFFT/FFT size efficient OFDM non-orthogonal multiple access (NOMA) for visible light communication,” IEEE Photonics J. 12(3), 1–5 (2020).
[Crossref]

Chow, C.-W.

Chowdhury, M. Z.

M. Z. Chowdhury, M. T. Hossan, A. Islam, and Y. M. Jang, “A comparative survey of optical wireless technologies: Architectures and applications,” IEEE Access 6, 9819–9840 (2018).
[Crossref]

Chu, W.

W. Chu, J. Dang, Z. Zhang, and L. Wu, “Effect of clipping on the achievable rate of non-orthogonal multiple access with DCO-OFDM,” in Proceedings of 9th International Conference on Wireless Communications and Signal Processing (2017), pp. 1–6.

Dai, L.

R. Jiang, Q. Wang, F. Wang, L. Dai, and Z. Wang, “An optimal scaling scheme for DCO-OFDM based visible light communications,” Opt. Commun. 356, 136–140 (2015).
[Crossref]

Dang, J.

L. Wu, Z. Zhang, J. Dang, and H. Liu, “Adaptive modulation schemes for visible light communications,” J. Lightwave Technol. 33(1), 117–125 (2015).
[Crossref]

W. Chu, J. Dang, Z. Zhang, and L. Wu, “Effect of clipping on the achievable rate of non-orthogonal multiple access with DCO-OFDM,” in Proceedings of 9th International Conference on Wireless Communications and Signal Processing (2017), pp. 1–6.

M. Yu, Z. Zhang, L. Wu, and J. Dang, “Optimal symmetric double-sided signal clipping in DCO-OFDM visible light communication,” in Proceedings of Sixth International Conference on Wireless Communications and Signal Processing (2014), pp. 1–4.

Dardari, D.

D. Dardari, V. Tralli, and A. Vaccari, “A theoretical characterization of nonlinear distortion effects in OFDM systems,” IEEE Trans. Commun. 48(10), 1755–1764 (2000).
[Crossref]

Dawson, M. D.

M. S. Islim, R. X. Ferreira, X. He, E. Xie, S. Videv, S. Viola, S. Watson, N. Bamiedakis, R. V. Penty, I. H. White, A. E. Kelly, E. Gu, H. Haas, and M. D. Dawson, “Towards 10 Gb/s orthogonal frequency division multiplexing-based visible light communication using a GaN violet micro-LED,” Photonics Res. 5(2), A35–A43 (2017).
[Crossref]

Dennis, J.

J. Dennis and R. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (SIAM, 1996).

Dimitrov, S.

S. Dimitrov and H. Haas, “Information rate of OFDM-based optical wireless communication systems with nonlinear distortion,” J. Lightwave Technol. 31(6), 918–929 (2013).
[Crossref]

S. Dimitrov, S. Sinanovic, and H. Haas, “Clipping noise in OFDM-based optical wireless communication systems,” IEEE Trans. Commun. 60(4), 1072–1081 (2012).
[Crossref]

S. Dimitrov, S. Sinanovic, and H. Haas, “Signal shaping and modulation for optical wireless communication,” J. Lightwave Technol. 30(9), 1319–1328 (2012).
[Crossref]

S. Dimitrov and H. Haas, “Optimum signal shaping in OFDM-based optical wireless communication systems,” in Proceedings of IEEE Vehicular Technology Conference (IEEE, 2012), pp. 1–5.

Ding, Z.

X. Ling, J. Wang, Z. Ding, C. Zhao, and X. Gao, “Efficient OFDMA for LiFi downlink,” J. Lightwave Technol. 36(10), 1928–1943 (2018).
[Crossref]

X. Ling, J. Wang, X. Liang, Z. Ding, C. Zhao, and X. Gao, “Biased multi-LED beamforming for multicarrier visible light communications,” IEEE J. Sel. Areas Commun. 36(1), 106–120 (2018).
[Crossref]

X. Ling, J. Wang, X. Liang, Z. Ding, and C. Zhao, “Offset and power optimization for DCO-OFDM in visible light communication systems,” IEEE Trans. Signal Process. 64(2), 349–363 (2016).
[Crossref]

J. Wang, Y. Xu, X. Ling, R. Zhang, Z. Ding, and C. Zhao, “PAPR analysis for OFDM visible light communication,” Opt. Express 24(24), 27457–27474 (2016).
[Crossref]

X. Ling, J. Wang, X. Liang, Z. Ding, and C. Zhao, “Joint offset and power optimization for visible light DCO-OFDM systems,” in Proceedings of IEEE Global Communications Conference (IEEE, 2015), pp. 1–6.

Dissanayake, S. D.

Elgala, H.

M. Ayyash, H. Elgala, A. Khreishah, V. Jungnickel, T. Little, S. Shao, M. Rahaim, D. Schulz, J. Hilt, and R. Freund, “Coexistence of WiFi and LiFi toward 5G: Concepts, opportunities, and challenges,” IEEE Commun. Mag. 54(2), 64–71 (2016).
[Crossref]

Evans, J.

L. Chen, B. Krongold, and J. Evans, “Theoretical characterization of nonlinear clipping effects in IM/DD optical OFDM systems,” IEEE Trans. Commun. 60(8), 2304–2312 (2012).
[Crossref]

Ferreira, R. X.

M. S. Islim, R. X. Ferreira, X. He, E. Xie, S. Videv, S. Viola, S. Watson, N. Bamiedakis, R. V. Penty, I. H. White, A. E. Kelly, E. Gu, H. Haas, and M. D. Dawson, “Towards 10 Gb/s orthogonal frequency division multiplexing-based visible light communication using a GaN violet micro-LED,” Photonics Res. 5(2), A35–A43 (2017).
[Crossref]

Freund, R.

M. Ayyash, H. Elgala, A. Khreishah, V. Jungnickel, T. Little, S. Shao, M. Rahaim, D. Schulz, J. Hilt, and R. Freund, “Coexistence of WiFi and LiFi toward 5G: Concepts, opportunities, and challenges,” IEEE Commun. Mag. 54(2), 64–71 (2016).
[Crossref]

Gao, X.

X. Ling, J. Wang, Z. Ding, C. Zhao, and X. Gao, “Efficient OFDMA for LiFi downlink,” J. Lightwave Technol. 36(10), 1928–1943 (2018).
[Crossref]

X. Ling, J. Wang, X. Liang, Z. Ding, C. Zhao, and X. Gao, “Biased multi-LED beamforming for multicarrier visible light communications,” IEEE J. Sel. Areas Commun. 36(1), 106–120 (2018).
[Crossref]

Ge, P.

P. Ge, J. Wang, X. Ling, X. Liang, Y. Tian, and C. Zhao, “Achievable rate analysis for post- and pre-equalization in DCO-OFDM VLC with limited dynamic range,” Opt. Commun. (to be published).

Gu, E.

M. S. Islim, R. X. Ferreira, X. He, E. Xie, S. Videv, S. Viola, S. Watson, N. Bamiedakis, R. V. Penty, I. H. White, A. E. Kelly, E. Gu, H. Haas, and M. D. Dawson, “Towards 10 Gb/s orthogonal frequency division multiplexing-based visible light communication using a GaN violet micro-LED,” Photonics Res. 5(2), A35–A43 (2017).
[Crossref]

Haas, H.

R. Bian, I. Tavakkolnia, and H. Haas, “15.73 Gb/s visible light communication with off-the-shelf LEDs,” J. Lightwave Technol. 37(10), 2418–2424 (2019).
[Crossref]

M. S. Islim, R. X. Ferreira, X. He, E. Xie, S. Videv, S. Viola, S. Watson, N. Bamiedakis, R. V. Penty, I. H. White, A. E. Kelly, E. Gu, H. Haas, and M. D. Dawson, “Towards 10 Gb/s orthogonal frequency division multiplexing-based visible light communication using a GaN violet micro-LED,” Photonics Res. 5(2), A35–A43 (2017).
[Crossref]

D. Tsonev, S. Sinanovic, and H. Haas, “Complete modeling of nonlinear distortion in OFDM-based optical wireless communication,” J. Lightwave Technol. 31(18), 3064–3076 (2013).
[Crossref]

S. Dimitrov and H. Haas, “Information rate of OFDM-based optical wireless communication systems with nonlinear distortion,” J. Lightwave Technol. 31(6), 918–929 (2013).
[Crossref]

S. Dimitrov, S. Sinanovic, and H. Haas, “Clipping noise in OFDM-based optical wireless communication systems,” IEEE Trans. Commun. 60(4), 1072–1081 (2012).
[Crossref]

S. Dimitrov, S. Sinanovic, and H. Haas, “Signal shaping and modulation for optical wireless communication,” J. Lightwave Technol. 30(9), 1319–1328 (2012).
[Crossref]

S. Dimitrov and H. Haas, “Optimum signal shaping in OFDM-based optical wireless communication systems,” in Proceedings of IEEE Vehicular Technology Conference (IEEE, 2012), pp. 1–5.

Hanzo, L.

X. Li, R. Zhang, and L. Hanzo, “Optimization of visible-light optical wireless systems: Network-centric versus user-centric designs,” IEEE Commun. Surv. Tutorials 20(3), 1878–1904 (2018).
[Crossref]

Z. Wang, Q. Wang, S. Chen, and L. Hanzo, “An adaptive scaling and biasing scheme for OFDM-based visible light communication systems,” Opt. Express 22(10), 12707–12715 (2014).
[Crossref]

He, X.

M. S. Islim, R. X. Ferreira, X. He, E. Xie, S. Videv, S. Viola, S. Watson, N. Bamiedakis, R. V. Penty, I. H. White, A. E. Kelly, E. Gu, H. Haas, and M. D. Dawson, “Towards 10 Gb/s orthogonal frequency division multiplexing-based visible light communication using a GaN violet micro-LED,” Photonics Res. 5(2), A35–A43 (2017).
[Crossref]

He, Y.

Y. He, M. Jiang, X. Ling, and C. Zhao, “Robust BICM design for the LDPC coded DCO-OFDM: A deep learning approach,” IEEE Trans. Commun. 68(2), 713–727 (2020).
[Crossref]

Y. He, M. Jiang, X. Ling, and C. Zhao, “Protograph-based EXIT analysis and optimization of LDPC coded DCO-OFDM in VLC systems,” IEEE Photonics Technol. Lett. 30(21), 1898–1901 (2018).
[Crossref]

Y. He, M. Jiang, X. Ling, and C. Zhao, “A neural network aided approach for LDPC coded DCO-OFDM with clipping distortion,” in Proceedings of IEEE International Conference on Communications (IEEE, 2019), pp. 1–6.

S. Ma, R. Yang, Y. He, S. Lu, F. Zhou, N. Al-Dhahir, and S. Li, “Achieving channel capacity of visible light communication,” IEEE Systems Journal pp. 1–12 (2020).

Hilt, J.

M. Ayyash, H. Elgala, A. Khreishah, V. Jungnickel, T. Little, S. Shao, M. Rahaim, D. Schulz, J. Hilt, and R. Freund, “Coexistence of WiFi and LiFi toward 5G: Concepts, opportunities, and challenges,” IEEE Commun. Mag. 54(2), 64–71 (2016).
[Crossref]

Hong, M.

M. Razaviyayn, M. Hong, and Z.-Q. Luo, “A unified convergence analysis of block successive minimization methods for nonsmooth optimization,” SIAM J. Optim. 23(2), 1126–1153 (2013).
[Crossref]

Hossan, M. T.

M. Z. Chowdhury, M. T. Hossan, A. Islam, and Y. M. Jang, “A comparative survey of optical wireless technologies: Architectures and applications,” IEEE Access 6, 9819–9840 (2018).
[Crossref]

Huang, N.

J. Cai, M. Chen, N. Huang, and J. Qian, “DC bias and power optimization for AV-DCO-OFDM in optical wireless communication,” Opt. Commun. (to be published).

Huang, X.

X. Huang, F. Yang, J. Song, and L. Cheng, “Subcarrier and power allocations for enhanced ADO-OFDM with dimming control,” in Proceedings of IEEE International Conference on Communications (IEEE, 2019), pp. 1–6.

Islam, A.

M. Z. Chowdhury, M. T. Hossan, A. Islam, and Y. M. Jang, “A comparative survey of optical wireless technologies: Architectures and applications,” IEEE Access 6, 9819–9840 (2018).
[Crossref]

Islim, M. S.

M. S. Islim, R. X. Ferreira, X. He, E. Xie, S. Videv, S. Viola, S. Watson, N. Bamiedakis, R. V. Penty, I. H. White, A. E. Kelly, E. Gu, H. Haas, and M. D. Dawson, “Towards 10 Gb/s orthogonal frequency division multiplexing-based visible light communication using a GaN violet micro-LED,” Photonics Res. 5(2), A35–A43 (2017).
[Crossref]

Jang, Y. M.

M. Z. Chowdhury, M. T. Hossan, A. Islam, and Y. M. Jang, “A comparative survey of optical wireless technologies: Architectures and applications,” IEEE Access 6, 9819–9840 (2018).
[Crossref]

Jiang, M.

Y. He, M. Jiang, X. Ling, and C. Zhao, “Robust BICM design for the LDPC coded DCO-OFDM: A deep learning approach,” IEEE Trans. Commun. 68(2), 713–727 (2020).
[Crossref]

Y. He, M. Jiang, X. Ling, and C. Zhao, “Protograph-based EXIT analysis and optimization of LDPC coded DCO-OFDM in VLC systems,” IEEE Photonics Technol. Lett. 30(21), 1898–1901 (2018).
[Crossref]

Y. He, M. Jiang, X. Ling, and C. Zhao, “A neural network aided approach for LDPC coded DCO-OFDM with clipping distortion,” in Proceedings of IEEE International Conference on Communications (IEEE, 2019), pp. 1–6.

Jiang, R.

R. Jiang, Q. Wang, F. Wang, L. Dai, and Z. Wang, “An optimal scaling scheme for DCO-OFDM based visible light communications,” Opt. Commun. 356, 136–140 (2015).
[Crossref]

Jungnickel, V.

M. Ayyash, H. Elgala, A. Khreishah, V. Jungnickel, T. Little, S. Shao, M. Rahaim, D. Schulz, J. Hilt, and R. Freund, “Coexistence of WiFi and LiFi toward 5G: Concepts, opportunities, and challenges,” IEEE Commun. Mag. 54(2), 64–71 (2016).
[Crossref]

Kahn, J. M.

J. R. Barry, J. M. Kahn, W. J. Krause, E. A. Lee, and D. G. Messerschmitt, “Simulation of multipath impulse response for indoor wireless optical channels,” IEEE J. Sel. Areas Commun. 11(3), 367–379 (1993).
[Crossref]

Kalavally, V.

D. Karunatilaka, F. Zafar, V. Kalavally, and R. Parthiban, “LED based indoor visible light communications: State of the art,” IEEE Commun. Surv. Tutorials 17(3), 1649–1678 (2015).
[Crossref]

Karunatilaka, D.

D. Karunatilaka, F. Zafar, V. Kalavally, and R. Parthiban, “LED based indoor visible light communications: State of the art,” IEEE Commun. Surv. Tutorials 17(3), 1649–1678 (2015).
[Crossref]

Kelly, A. E.

M. S. Islim, R. X. Ferreira, X. He, E. Xie, S. Videv, S. Viola, S. Watson, N. Bamiedakis, R. V. Penty, I. H. White, A. E. Kelly, E. Gu, H. Haas, and M. D. Dawson, “Towards 10 Gb/s orthogonal frequency division multiplexing-based visible light communication using a GaN violet micro-LED,” Photonics Res. 5(2), A35–A43 (2017).
[Crossref]

Khreishah, A.

M. Ayyash, H. Elgala, A. Khreishah, V. Jungnickel, T. Little, S. Shao, M. Rahaim, D. Schulz, J. Hilt, and R. Freund, “Coexistence of WiFi and LiFi toward 5G: Concepts, opportunities, and challenges,” IEEE Commun. Mag. 54(2), 64–71 (2016).
[Crossref]

Krause, W. J.

J. R. Barry, J. M. Kahn, W. J. Krause, E. A. Lee, and D. G. Messerschmitt, “Simulation of multipath impulse response for indoor wireless optical channels,” IEEE J. Sel. Areas Commun. 11(3), 367–379 (1993).
[Crossref]

Krongold, B.

L. Chen, B. Krongold, and J. Evans, “Theoretical characterization of nonlinear clipping effects in IM/DD optical OFDM systems,” IEEE Trans. Commun. 60(8), 2304–2312 (2012).
[Crossref]

Lee, E. A.

J. R. Barry, J. M. Kahn, W. J. Krause, E. A. Lee, and D. G. Messerschmitt, “Simulation of multipath impulse response for indoor wireless optical channels,” IEEE J. Sel. Areas Commun. 11(3), 367–379 (1993).
[Crossref]

Lee, K.

K. Lee, H. Park, and J. R. Barry, “Indoor channel characteristics for visible light communications,” IEEE Commun. Lett. 15(2), 217–219 (2011).
[Crossref]

Li, S.

S. Ma, R. Yang, Y. He, S. Lu, F. Zhou, N. Al-Dhahir, and S. Li, “Achieving channel capacity of visible light communication,” IEEE Systems Journal pp. 1–12 (2020).

Li, X.

X. Li, R. Zhang, and L. Hanzo, “Optimization of visible-light optical wireless systems: Network-centric versus user-centric designs,” IEEE Commun. Surv. Tutorials 20(3), 1878–1904 (2018).
[Crossref]

Liang, X.

X. Ling, J. Wang, X. Liang, Z. Ding, C. Zhao, and X. Gao, “Biased multi-LED beamforming for multicarrier visible light communications,” IEEE J. Sel. Areas Commun. 36(1), 106–120 (2018).
[Crossref]

X. Ling, J. Wang, X. Liang, Z. Ding, and C. Zhao, “Offset and power optimization for DCO-OFDM in visible light communication systems,” IEEE Trans. Signal Process. 64(2), 349–363 (2016).
[Crossref]

P. Ge, J. Wang, X. Ling, X. Liang, Y. Tian, and C. Zhao, “Achievable rate analysis for post- and pre-equalization in DCO-OFDM VLC with limited dynamic range,” Opt. Commun. (to be published).

X. Ling, J. Wang, X. Liang, Z. Ding, and C. Zhao, “Joint offset and power optimization for visible light DCO-OFDM systems,” in Proceedings of IEEE Global Communications Conference (IEEE, 2015), pp. 1–6.

Ling, X.

Y. He, M. Jiang, X. Ling, and C. Zhao, “Robust BICM design for the LDPC coded DCO-OFDM: A deep learning approach,” IEEE Trans. Commun. 68(2), 713–727 (2020).
[Crossref]

X. Ling, J. Wang, Z. Ding, C. Zhao, and X. Gao, “Efficient OFDMA for LiFi downlink,” J. Lightwave Technol. 36(10), 1928–1943 (2018).
[Crossref]

Y. He, M. Jiang, X. Ling, and C. Zhao, “Protograph-based EXIT analysis and optimization of LDPC coded DCO-OFDM in VLC systems,” IEEE Photonics Technol. Lett. 30(21), 1898–1901 (2018).
[Crossref]

X. Ling, J. Wang, X. Liang, Z. Ding, C. Zhao, and X. Gao, “Biased multi-LED beamforming for multicarrier visible light communications,” IEEE J. Sel. Areas Commun. 36(1), 106–120 (2018).
[Crossref]

X. Ling, J. Wang, X. Liang, Z. Ding, and C. Zhao, “Offset and power optimization for DCO-OFDM in visible light communication systems,” IEEE Trans. Signal Process. 64(2), 349–363 (2016).
[Crossref]

J. Wang, Y. Xu, X. Ling, R. Zhang, Z. Ding, and C. Zhao, “PAPR analysis for OFDM visible light communication,” Opt. Express 24(24), 27457–27474 (2016).
[Crossref]

P. Ge, J. Wang, X. Ling, X. Liang, Y. Tian, and C. Zhao, “Achievable rate analysis for post- and pre-equalization in DCO-OFDM VLC with limited dynamic range,” Opt. Commun. (to be published).

X. Ling, J. Wang, X. Liang, Z. Ding, and C. Zhao, “Joint offset and power optimization for visible light DCO-OFDM systems,” in Proceedings of IEEE Global Communications Conference (IEEE, 2015), pp. 1–6.

Y. He, M. Jiang, X. Ling, and C. Zhao, “A neural network aided approach for LDPC coded DCO-OFDM with clipping distortion,” in Proceedings of IEEE International Conference on Communications (IEEE, 2019), pp. 1–6.

Little, T.

M. Ayyash, H. Elgala, A. Khreishah, V. Jungnickel, T. Little, S. Shao, M. Rahaim, D. Schulz, J. Hilt, and R. Freund, “Coexistence of WiFi and LiFi toward 5G: Concepts, opportunities, and challenges,” IEEE Commun. Mag. 54(2), 64–71 (2016).
[Crossref]

Liu, H.

Liu, P.

Z. Lu, P. Liu, and S. Liu, “Experiment of beneficial clipping DCO-OFDM based visible light communication,” in Proceedings of IEEE 9th International Conference on Communication Software and Networks (IEEE, 2017), pp. 615–618.

Liu, S.

Z. Lu, P. Liu, and S. Liu, “Experiment of beneficial clipping DCO-OFDM based visible light communication,” in Proceedings of IEEE 9th International Conference on Communication Software and Networks (IEEE, 2017), pp. 615–618.

Liu, Y.

A. Adnan, Y. Liu, C.-W. Chow, and C.-H. Yeh, “Demonstration of non-Hermitian symmetry (NHS) serial-complex-valued orthogonal frequency division multiplexing (SCV-OFDM) for white-light visible light communication (VLC),” OSA Continuum 3(5), 1163–1168 (2020).
[Crossref]

A. Adnan, Y. Liu, C. Chow, and C. Yeh, “Demonstration of non-Hermitian symmetry (NHS) IFFT/FFT size efficient OFDM non-orthogonal multiple access (NOMA) for visible light communication,” IEEE Photonics J. 12(3), 1–5 (2020).
[Crossref]

Liu, Y.-L.

Lu, S.

S. Ma, R. Yang, Y. He, S. Lu, F. Zhou, N. Al-Dhahir, and S. Li, “Achieving channel capacity of visible light communication,” IEEE Systems Journal pp. 1–12 (2020).

Lu, Z.

Z. Lu, P. Liu, and S. Liu, “Experiment of beneficial clipping DCO-OFDM based visible light communication,” in Proceedings of IEEE 9th International Conference on Communication Software and Networks (IEEE, 2017), pp. 615–618.

Luo, Z.-Q.

M. Razaviyayn, M. Hong, and Z.-Q. Luo, “A unified convergence analysis of block successive minimization methods for nonsmooth optimization,” SIAM J. Optim. 23(2), 1126–1153 (2013).
[Crossref]

Ma, S.

S. Ma, R. Yang, Y. He, S. Lu, F. Zhou, N. Al-Dhahir, and S. Li, “Achieving channel capacity of visible light communication,” IEEE Systems Journal pp. 1–12 (2020).

Messerschmitt, D. G.

J. R. Barry, J. M. Kahn, W. J. Krause, E. A. Lee, and D. G. Messerschmitt, “Simulation of multipath impulse response for indoor wireless optical channels,” IEEE J. Sel. Areas Commun. 11(3), 367–379 (1993).
[Crossref]

Mogensen, P. E.

H. Viswanathan and P. E. Mogensen, “Communications in the 6G era,” IEEE Access 8, 57063–57074 (2020).
[Crossref]

Oppenheim, A. V.

A. V. Oppenheim, Discrete-Time Signal Processing (Pearson Education India, 1999).

Park, H.

K. Lee, H. Park, and J. R. Barry, “Indoor channel characteristics for visible light communications,” IEEE Commun. Lett. 15(2), 217–219 (2011).
[Crossref]

Parthiban, R.

D. Karunatilaka, F. Zafar, V. Kalavally, and R. Parthiban, “LED based indoor visible light communications: State of the art,” IEEE Commun. Surv. Tutorials 17(3), 1649–1678 (2015).
[Crossref]

Penty, R. V.

M. S. Islim, R. X. Ferreira, X. He, E. Xie, S. Videv, S. Viola, S. Watson, N. Bamiedakis, R. V. Penty, I. H. White, A. E. Kelly, E. Gu, H. Haas, and M. D. Dawson, “Towards 10 Gb/s orthogonal frequency division multiplexing-based visible light communication using a GaN violet micro-LED,” Photonics Res. 5(2), A35–A43 (2017).
[Crossref]

Qian, J.

J. Cai, M. Chen, N. Huang, and J. Qian, “DC bias and power optimization for AV-DCO-OFDM in optical wireless communication,” Opt. Commun. (to be published).

Rahaim, M.

M. Ayyash, H. Elgala, A. Khreishah, V. Jungnickel, T. Little, S. Shao, M. Rahaim, D. Schulz, J. Hilt, and R. Freund, “Coexistence of WiFi and LiFi toward 5G: Concepts, opportunities, and challenges,” IEEE Commun. Mag. 54(2), 64–71 (2016).
[Crossref]

Razaviyayn, M.

M. Razaviyayn, M. Hong, and Z.-Q. Luo, “A unified convergence analysis of block successive minimization methods for nonsmooth optimization,” SIAM J. Optim. 23(2), 1126–1153 (2013).
[Crossref]

Schnabel, R.

J. Dennis and R. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (SIAM, 1996).

Schulz, D.

M. Ayyash, H. Elgala, A. Khreishah, V. Jungnickel, T. Little, S. Shao, M. Rahaim, D. Schulz, J. Hilt, and R. Freund, “Coexistence of WiFi and LiFi toward 5G: Concepts, opportunities, and challenges,” IEEE Commun. Mag. 54(2), 64–71 (2016).
[Crossref]

Shao, S.

M. Ayyash, H. Elgala, A. Khreishah, V. Jungnickel, T. Little, S. Shao, M. Rahaim, D. Schulz, J. Hilt, and R. Freund, “Coexistence of WiFi and LiFi toward 5G: Concepts, opportunities, and challenges,” IEEE Commun. Mag. 54(2), 64–71 (2016).
[Crossref]

Sinanovic, S.

Song, J.

X. Huang, F. Yang, J. Song, and L. Cheng, “Subcarrier and power allocations for enhanced ADO-OFDM with dimming control,” in Proceedings of IEEE International Conference on Communications (IEEE, 2019), pp. 1–6.

Tavakkolnia, I.

Tian, Y.

P. Ge, J. Wang, X. Ling, X. Liang, Y. Tian, and C. Zhao, “Achievable rate analysis for post- and pre-equalization in DCO-OFDM VLC with limited dynamic range,” Opt. Commun. (to be published).

Tralli, V.

D. Dardari, V. Tralli, and A. Vaccari, “A theoretical characterization of nonlinear distortion effects in OFDM systems,” IEEE Trans. Commun. 48(10), 1755–1764 (2000).
[Crossref]

Tsonev, D.

Vaccari, A.

D. Dardari, V. Tralli, and A. Vaccari, “A theoretical characterization of nonlinear distortion effects in OFDM systems,” IEEE Trans. Commun. 48(10), 1755–1764 (2000).
[Crossref]

Videv, S.

M. S. Islim, R. X. Ferreira, X. He, E. Xie, S. Videv, S. Viola, S. Watson, N. Bamiedakis, R. V. Penty, I. H. White, A. E. Kelly, E. Gu, H. Haas, and M. D. Dawson, “Towards 10 Gb/s orthogonal frequency division multiplexing-based visible light communication using a GaN violet micro-LED,” Photonics Res. 5(2), A35–A43 (2017).
[Crossref]

Viola, S.

M. S. Islim, R. X. Ferreira, X. He, E. Xie, S. Videv, S. Viola, S. Watson, N. Bamiedakis, R. V. Penty, I. H. White, A. E. Kelly, E. Gu, H. Haas, and M. D. Dawson, “Towards 10 Gb/s orthogonal frequency division multiplexing-based visible light communication using a GaN violet micro-LED,” Photonics Res. 5(2), A35–A43 (2017).
[Crossref]

Viswanathan, H.

H. Viswanathan and P. E. Mogensen, “Communications in the 6G era,” IEEE Access 8, 57063–57074 (2020).
[Crossref]

Wang, F.

R. Jiang, Q. Wang, F. Wang, L. Dai, and Z. Wang, “An optimal scaling scheme for DCO-OFDM based visible light communications,” Opt. Commun. 356, 136–140 (2015).
[Crossref]

Wang, J.

X. Ling, J. Wang, Z. Ding, C. Zhao, and X. Gao, “Efficient OFDMA for LiFi downlink,” J. Lightwave Technol. 36(10), 1928–1943 (2018).
[Crossref]

X. Ling, J. Wang, X. Liang, Z. Ding, C. Zhao, and X. Gao, “Biased multi-LED beamforming for multicarrier visible light communications,” IEEE J. Sel. Areas Commun. 36(1), 106–120 (2018).
[Crossref]

X. Ling, J. Wang, X. Liang, Z. Ding, and C. Zhao, “Offset and power optimization for DCO-OFDM in visible light communication systems,” IEEE Trans. Signal Process. 64(2), 349–363 (2016).
[Crossref]

J. Wang, Y. Xu, X. Ling, R. Zhang, Z. Ding, and C. Zhao, “PAPR analysis for OFDM visible light communication,” Opt. Express 24(24), 27457–27474 (2016).
[Crossref]

P. Ge, J. Wang, X. Ling, X. Liang, Y. Tian, and C. Zhao, “Achievable rate analysis for post- and pre-equalization in DCO-OFDM VLC with limited dynamic range,” Opt. Commun. (to be published).

X. Ling, J. Wang, X. Liang, Z. Ding, and C. Zhao, “Joint offset and power optimization for visible light DCO-OFDM systems,” in Proceedings of IEEE Global Communications Conference (IEEE, 2015), pp. 1–6.

Wang, Q.

R. Jiang, Q. Wang, F. Wang, L. Dai, and Z. Wang, “An optimal scaling scheme for DCO-OFDM based visible light communications,” Opt. Commun. 356, 136–140 (2015).
[Crossref]

Z. Wang, Q. Wang, S. Chen, and L. Hanzo, “An adaptive scaling and biasing scheme for OFDM-based visible light communication systems,” Opt. Express 22(10), 12707–12715 (2014).
[Crossref]

Wang, Z.

J. Wu and Z. Wang, “Theoretical and experimental investigation of direct detection optical OFDM systems with clipping and normalization,” J. Electr. Comput. Eng. 2018, 1–11 (2018).
[Crossref]

R. Jiang, Q. Wang, F. Wang, L. Dai, and Z. Wang, “An optimal scaling scheme for DCO-OFDM based visible light communications,” Opt. Commun. 356, 136–140 (2015).
[Crossref]

Z. Wang, Q. Wang, S. Chen, and L. Hanzo, “An adaptive scaling and biasing scheme for OFDM-based visible light communication systems,” Opt. Express 22(10), 12707–12715 (2014).
[Crossref]

Watson, S.

M. S. Islim, R. X. Ferreira, X. He, E. Xie, S. Videv, S. Viola, S. Watson, N. Bamiedakis, R. V. Penty, I. H. White, A. E. Kelly, E. Gu, H. Haas, and M. D. Dawson, “Towards 10 Gb/s orthogonal frequency division multiplexing-based visible light communication using a GaN violet micro-LED,” Photonics Res. 5(2), A35–A43 (2017).
[Crossref]

White, I. H.

M. S. Islim, R. X. Ferreira, X. He, E. Xie, S. Videv, S. Viola, S. Watson, N. Bamiedakis, R. V. Penty, I. H. White, A. E. Kelly, E. Gu, H. Haas, and M. D. Dawson, “Towards 10 Gb/s orthogonal frequency division multiplexing-based visible light communication using a GaN violet micro-LED,” Photonics Res. 5(2), A35–A43 (2017).
[Crossref]

Wu, J.

J. Wu and Z. Wang, “Theoretical and experimental investigation of direct detection optical OFDM systems with clipping and normalization,” J. Electr. Comput. Eng. 2018, 1–11 (2018).
[Crossref]

Wu, L.

L. Wu, Z. Zhang, J. Dang, and H. Liu, “Adaptive modulation schemes for visible light communications,” J. Lightwave Technol. 33(1), 117–125 (2015).
[Crossref]

M. Yu, Z. Zhang, L. Wu, and J. Dang, “Optimal symmetric double-sided signal clipping in DCO-OFDM visible light communication,” in Proceedings of Sixth International Conference on Wireless Communications and Signal Processing (2014), pp. 1–4.

W. Chu, J. Dang, Z. Zhang, and L. Wu, “Effect of clipping on the achievable rate of non-orthogonal multiple access with DCO-OFDM,” in Proceedings of 9th International Conference on Wireless Communications and Signal Processing (2017), pp. 1–6.

Xie, E.

M. S. Islim, R. X. Ferreira, X. He, E. Xie, S. Videv, S. Viola, S. Watson, N. Bamiedakis, R. V. Penty, I. H. White, A. E. Kelly, E. Gu, H. Haas, and M. D. Dawson, “Towards 10 Gb/s orthogonal frequency division multiplexing-based visible light communication using a GaN violet micro-LED,” Photonics Res. 5(2), A35–A43 (2017).
[Crossref]

Xu, Y.

Yang, F.

X. Huang, F. Yang, J. Song, and L. Cheng, “Subcarrier and power allocations for enhanced ADO-OFDM with dimming control,” in Proceedings of IEEE International Conference on Communications (IEEE, 2019), pp. 1–6.

Yang, R.

S. Ma, R. Yang, Y. He, S. Lu, F. Zhou, N. Al-Dhahir, and S. Li, “Achieving channel capacity of visible light communication,” IEEE Systems Journal pp. 1–12 (2020).

Yeh, C.

A. Adnan, Y. Liu, C. Chow, and C. Yeh, “Demonstration of non-Hermitian symmetry (NHS) IFFT/FFT size efficient OFDM non-orthogonal multiple access (NOMA) for visible light communication,” IEEE Photonics J. 12(3), 1–5 (2020).
[Crossref]

Yeh, C.-H.

Yu, M.

M. Yu, Z. Zhang, L. Wu, and J. Dang, “Optimal symmetric double-sided signal clipping in DCO-OFDM visible light communication,” in Proceedings of Sixth International Conference on Wireless Communications and Signal Processing (2014), pp. 1–4.

Zafar, F.

D. Karunatilaka, F. Zafar, V. Kalavally, and R. Parthiban, “LED based indoor visible light communications: State of the art,” IEEE Commun. Surv. Tutorials 17(3), 1649–1678 (2015).
[Crossref]

Zhang, M.

M. Zhang and Z. Zhang, “An optimum DC-biasing for DCO-OFDM system,” IEEE Commun. Lett. 18(8), 1351–1354 (2014).
[Crossref]

Zhang, R.

X. Li, R. Zhang, and L. Hanzo, “Optimization of visible-light optical wireless systems: Network-centric versus user-centric designs,” IEEE Commun. Surv. Tutorials 20(3), 1878–1904 (2018).
[Crossref]

J. Wang, Y. Xu, X. Ling, R. Zhang, Z. Ding, and C. Zhao, “PAPR analysis for OFDM visible light communication,” Opt. Express 24(24), 27457–27474 (2016).
[Crossref]

Zhang, Z.

L. Wu, Z. Zhang, J. Dang, and H. Liu, “Adaptive modulation schemes for visible light communications,” J. Lightwave Technol. 33(1), 117–125 (2015).
[Crossref]

M. Zhang and Z. Zhang, “An optimum DC-biasing for DCO-OFDM system,” IEEE Commun. Lett. 18(8), 1351–1354 (2014).
[Crossref]

M. Yu, Z. Zhang, L. Wu, and J. Dang, “Optimal symmetric double-sided signal clipping in DCO-OFDM visible light communication,” in Proceedings of Sixth International Conference on Wireless Communications and Signal Processing (2014), pp. 1–4.

W. Chu, J. Dang, Z. Zhang, and L. Wu, “Effect of clipping on the achievable rate of non-orthogonal multiple access with DCO-OFDM,” in Proceedings of 9th International Conference on Wireless Communications and Signal Processing (2017), pp. 1–6.

Zhao, C.

Y. He, M. Jiang, X. Ling, and C. Zhao, “Robust BICM design for the LDPC coded DCO-OFDM: A deep learning approach,” IEEE Trans. Commun. 68(2), 713–727 (2020).
[Crossref]

X. Ling, J. Wang, Z. Ding, C. Zhao, and X. Gao, “Efficient OFDMA for LiFi downlink,” J. Lightwave Technol. 36(10), 1928–1943 (2018).
[Crossref]

Y. He, M. Jiang, X. Ling, and C. Zhao, “Protograph-based EXIT analysis and optimization of LDPC coded DCO-OFDM in VLC systems,” IEEE Photonics Technol. Lett. 30(21), 1898–1901 (2018).
[Crossref]

X. Ling, J. Wang, X. Liang, Z. Ding, C. Zhao, and X. Gao, “Biased multi-LED beamforming for multicarrier visible light communications,” IEEE J. Sel. Areas Commun. 36(1), 106–120 (2018).
[Crossref]

X. Ling, J. Wang, X. Liang, Z. Ding, and C. Zhao, “Offset and power optimization for DCO-OFDM in visible light communication systems,” IEEE Trans. Signal Process. 64(2), 349–363 (2016).
[Crossref]

J. Wang, Y. Xu, X. Ling, R. Zhang, Z. Ding, and C. Zhao, “PAPR analysis for OFDM visible light communication,” Opt. Express 24(24), 27457–27474 (2016).
[Crossref]

P. Ge, J. Wang, X. Ling, X. Liang, Y. Tian, and C. Zhao, “Achievable rate analysis for post- and pre-equalization in DCO-OFDM VLC with limited dynamic range,” Opt. Commun. (to be published).

X. Ling, J. Wang, X. Liang, Z. Ding, and C. Zhao, “Joint offset and power optimization for visible light DCO-OFDM systems,” in Proceedings of IEEE Global Communications Conference (IEEE, 2015), pp. 1–6.

Y. He, M. Jiang, X. Ling, and C. Zhao, “A neural network aided approach for LDPC coded DCO-OFDM with clipping distortion,” in Proceedings of IEEE International Conference on Communications (IEEE, 2019), pp. 1–6.

Zhou, F.

S. Ma, R. Yang, Y. He, S. Lu, F. Zhou, N. Al-Dhahir, and S. Li, “Achieving channel capacity of visible light communication,” IEEE Systems Journal pp. 1–12 (2020).

IEEE Access (2)

H. Viswanathan and P. E. Mogensen, “Communications in the 6G era,” IEEE Access 8, 57063–57074 (2020).
[Crossref]

M. Z. Chowdhury, M. T. Hossan, A. Islam, and Y. M. Jang, “A comparative survey of optical wireless technologies: Architectures and applications,” IEEE Access 6, 9819–9840 (2018).
[Crossref]

IEEE Commun. Lett. (2)

M. Zhang and Z. Zhang, “An optimum DC-biasing for DCO-OFDM system,” IEEE Commun. Lett. 18(8), 1351–1354 (2014).
[Crossref]

K. Lee, H. Park, and J. R. Barry, “Indoor channel characteristics for visible light communications,” IEEE Commun. Lett. 15(2), 217–219 (2011).
[Crossref]

IEEE Commun. Mag. (1)

M. Ayyash, H. Elgala, A. Khreishah, V. Jungnickel, T. Little, S. Shao, M. Rahaim, D. Schulz, J. Hilt, and R. Freund, “Coexistence of WiFi and LiFi toward 5G: Concepts, opportunities, and challenges,” IEEE Commun. Mag. 54(2), 64–71 (2016).
[Crossref]

IEEE Commun. Surv. Tutorials (2)

X. Li, R. Zhang, and L. Hanzo, “Optimization of visible-light optical wireless systems: Network-centric versus user-centric designs,” IEEE Commun. Surv. Tutorials 20(3), 1878–1904 (2018).
[Crossref]

D. Karunatilaka, F. Zafar, V. Kalavally, and R. Parthiban, “LED based indoor visible light communications: State of the art,” IEEE Commun. Surv. Tutorials 17(3), 1649–1678 (2015).
[Crossref]

IEEE J. Sel. Areas Commun. (2)

X. Ling, J. Wang, X. Liang, Z. Ding, C. Zhao, and X. Gao, “Biased multi-LED beamforming for multicarrier visible light communications,” IEEE J. Sel. Areas Commun. 36(1), 106–120 (2018).
[Crossref]

J. R. Barry, J. M. Kahn, W. J. Krause, E. A. Lee, and D. G. Messerschmitt, “Simulation of multipath impulse response for indoor wireless optical channels,” IEEE J. Sel. Areas Commun. 11(3), 367–379 (1993).
[Crossref]

IEEE Photonics J. (1)

A. Adnan, Y. Liu, C. Chow, and C. Yeh, “Demonstration of non-Hermitian symmetry (NHS) IFFT/FFT size efficient OFDM non-orthogonal multiple access (NOMA) for visible light communication,” IEEE Photonics J. 12(3), 1–5 (2020).
[Crossref]

IEEE Photonics Technol. Lett. (1)

Y. He, M. Jiang, X. Ling, and C. Zhao, “Protograph-based EXIT analysis and optimization of LDPC coded DCO-OFDM in VLC systems,” IEEE Photonics Technol. Lett. 30(21), 1898–1901 (2018).
[Crossref]

IEEE Trans. Commun. (4)

L. Chen, B. Krongold, and J. Evans, “Theoretical characterization of nonlinear clipping effects in IM/DD optical OFDM systems,” IEEE Trans. Commun. 60(8), 2304–2312 (2012).
[Crossref]

D. Dardari, V. Tralli, and A. Vaccari, “A theoretical characterization of nonlinear distortion effects in OFDM systems,” IEEE Trans. Commun. 48(10), 1755–1764 (2000).
[Crossref]

Y. He, M. Jiang, X. Ling, and C. Zhao, “Robust BICM design for the LDPC coded DCO-OFDM: A deep learning approach,” IEEE Trans. Commun. 68(2), 713–727 (2020).
[Crossref]

S. Dimitrov, S. Sinanovic, and H. Haas, “Clipping noise in OFDM-based optical wireless communication systems,” IEEE Trans. Commun. 60(4), 1072–1081 (2012).
[Crossref]

IEEE Trans. Signal Process. (1)

X. Ling, J. Wang, X. Liang, Z. Ding, and C. Zhao, “Offset and power optimization for DCO-OFDM in visible light communication systems,” IEEE Trans. Signal Process. 64(2), 349–363 (2016).
[Crossref]

J. Electr. Comput. Eng. (1)

J. Wu and Z. Wang, “Theoretical and experimental investigation of direct detection optical OFDM systems with clipping and normalization,” J. Electr. Comput. Eng. 2018, 1–11 (2018).
[Crossref]

J. Lightwave Technol. (8)

Opt. Commun. (1)

R. Jiang, Q. Wang, F. Wang, L. Dai, and Z. Wang, “An optimal scaling scheme for DCO-OFDM based visible light communications,” Opt. Commun. 356, 136–140 (2015).
[Crossref]

Opt. Express (3)

OSA Continuum (1)

Photonics Res. (1)

M. S. Islim, R. X. Ferreira, X. He, E. Xie, S. Videv, S. Viola, S. Watson, N. Bamiedakis, R. V. Penty, I. H. White, A. E. Kelly, E. Gu, H. Haas, and M. D. Dawson, “Towards 10 Gb/s orthogonal frequency division multiplexing-based visible light communication using a GaN violet micro-LED,” Photonics Res. 5(2), A35–A43 (2017).
[Crossref]

SIAM J. Optim. (1)

M. Razaviyayn, M. Hong, and Z.-Q. Luo, “A unified convergence analysis of block successive minimization methods for nonsmooth optimization,” SIAM J. Optim. 23(2), 1126–1153 (2013).
[Crossref]

Other (14)

OSRAM GmbH, “Datasheet: LE UW S2LN,” (2011).

LED Engin, “Datasheet: LZ4-00MA00 RGBA LED,” (2016).

P. Ge, J. Wang, X. Ling, X. Liang, Y. Tian, and C. Zhao, “Achievable rate analysis for post- and pre-equalization in DCO-OFDM VLC with limited dynamic range,” Opt. Commun. (to be published).

Z. Lu, P. Liu, and S. Liu, “Experiment of beneficial clipping DCO-OFDM based visible light communication,” in Proceedings of IEEE 9th International Conference on Communication Software and Networks (IEEE, 2017), pp. 615–618.

S. Ma, R. Yang, Y. He, S. Lu, F. Zhou, N. Al-Dhahir, and S. Li, “Achieving channel capacity of visible light communication,” IEEE Systems Journal pp. 1–12 (2020).

S. Dimitrov and H. Haas, “Optimum signal shaping in OFDM-based optical wireless communication systems,” in Proceedings of IEEE Vehicular Technology Conference (IEEE, 2012), pp. 1–5.

M. Yu, Z. Zhang, L. Wu, and J. Dang, “Optimal symmetric double-sided signal clipping in DCO-OFDM visible light communication,” in Proceedings of Sixth International Conference on Wireless Communications and Signal Processing (2014), pp. 1–4.

J. Cai, M. Chen, N. Huang, and J. Qian, “DC bias and power optimization for AV-DCO-OFDM in optical wireless communication,” Opt. Commun. (to be published).

W. Chu, J. Dang, Z. Zhang, and L. Wu, “Effect of clipping on the achievable rate of non-orthogonal multiple access with DCO-OFDM,” in Proceedings of 9th International Conference on Wireless Communications and Signal Processing (2017), pp. 1–6.

X. Huang, F. Yang, J. Song, and L. Cheng, “Subcarrier and power allocations for enhanced ADO-OFDM with dimming control,” in Proceedings of IEEE International Conference on Communications (IEEE, 2019), pp. 1–6.

X. Ling, J. Wang, X. Liang, Z. Ding, and C. Zhao, “Joint offset and power optimization for visible light DCO-OFDM systems,” in Proceedings of IEEE Global Communications Conference (IEEE, 2015), pp. 1–6.

Y. He, M. Jiang, X. Ling, and C. Zhao, “A neural network aided approach for LDPC coded DCO-OFDM with clipping distortion,” in Proceedings of IEEE International Conference on Communications (IEEE, 2019), pp. 1–6.

A. V. Oppenheim, Discrete-Time Signal Processing (Pearson Education India, 1999).

J. Dennis and R. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (SIAM, 1996).

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Figures (7)

Fig. 1.
Fig. 1. Double-sided clipping DCO-OFDM system diagram.
Fig. 2.
Fig. 2. Numerical illustrations for bias optimization. (a) $a\left (b,p_{s}\right )$ and $p_{c}\left (b,p_{s}\right )$. (b) $r\left (b,\left \{ p_{k}\right \} \right )/\left (2N\right )$ and ${\cal I}\left (y_{n};s_{clip,n}|b,\left \{ p_{k}\right \} \right )$. (Configurations: $A=$700, $\sqrt {p_{k}}=$300, $\sqrt {p_{n}}=$30, $H_{k}=1$ and $2N=512$.)
Fig. 3.
Fig. 3. The optimal scheme and benchmarks over flat channels. (a) Achievable rate. (b) Signal shaping parameters.
Fig. 4.
Fig. 4. The optimal scheme and benchmarks over dispersive channels. (a) Achievable rate. (b) Convergence process of the optimal scheme ($\gamma =20$dB).
Fig. 5.
Fig. 5. Equivalent relationships between double-sided and single-sided clipping DCO-OFDM. (a) Optimal bias $b^{\star }$. (b) Optimal power $p_{s}^{\star }$.
Fig. 6.
Fig. 6. BER performance for different signal shaping schemes. (a) Uncoded DCO-OFDM. (b) Convolutional-coded DCO-OFDM. (c) Linear-coded DCO-OFDM.
Fig. 7.
Fig. 7. The constellation diagrams of different signal shaping schemes for uncoded DCO-OFDM at optical SNR $\gamma =25$dB. (a) Optimal. (b) Single-sided. (c) Approximated. (d) Adaptive. (e) $b=A/2,\sqrt {p_{s}}=A/8$. (f) $b=A/4,\sqrt {p_{s}}=A/8$.

Tables (1)

Tables Icon

Table 1. Parameters for simulations.

Equations (28)

Equations on this page are rendered with MathJax. Learn more.

s c l i p , n = clip [ s n + b ; A ] = { A s n + b A s n + b 0 < s n + b < A 0 s n + b 0 ,
f s c l i p , n ( w ; b , p s ) = 1 p s ϕ ( w b p s ) ( u ( w ) u ( w A ) ) + Q ( A b p s ) δ ( w A ) + Q ( b p s ) δ ( w ) ,
Y k = H k S c l i p , k + N k ,
S c l i p , k = a S k + N c l i p , k 1 k N 1 ,
p c ( b , p s ) p s ( Q ( b p s ) Q ( A b p s ) ) ( Q ( b p s ) + Q ( A b p s ) ) ( p s ϕ ( b p s ) p s ϕ ( A b p s ) b Q ( b p s ) + ( A b ) Q ( A b p s ) ) 2 + b 2 Q ( b p s ) + ( A b ) 2 Q ( A b p s ) b p s ϕ ( b p s ) ( A b ) p s ϕ ( A b p s ) .
Y k = a H k S k + H k N c l i p , k + N k , 1 k N 1.
r ( b , { p k } ) k = 1 N 1 log ( 1 + p k p s SNDR k ( b , p s ) ) ,
SNDR k ( b , p s ) | H k | 2 a 2 ( b , p s ) p s | H k | 2 p c ( b , p s ) + p n .
P : maximize b , { p k } r ( b , { p k } ) = k = 1 N 1 log ( 1 + p k p s SNDR k ( b , p s ) ) subject to k = 1 N 1 p k / N = p s .
b a ( b , p s ) { > 0 , 0 < b < A / 2 < 0 , A / 2 < b < A .
I ( y n ; s c l i p , n | b , { p k } ) = I ( y n ; s c l i p , n | A b , { p k } ) ,
s c l i p , n clip [ s n + b ; A ] , s c l i p , n clip [ A b s n ; A ] .
s c l i p , n = clip [ A b s n ; A ] = A clip [ s n + b ; A ] = A s c l i p , n .
I ( y n ; s c l i p , n | b , { p k } ) = H ( y n ) H ( y n | s c l i p , n ) = H ( y n ) H ( y n | s c l i p , n ) = I ( y n ; s c l i p , n | A b , { p k } ) .
d d p s SNDR k ( A / 2 , p s ) = A SNDR k 2 ( A / 2 , p s ) a 3 p s 3 / 2 q ( p s ) f ( p s ; γ ) ,
lim p s 0 + f ( p s ; γ ) = A 4 γ p s < 0 , lim p s + f ( p s ; γ ) = ϕ ( A 2 p s ) > 0 , and , d d p s f ( p s ; γ ) = A 4 p s 3 / 2 Q ( A 2 p s ) + A 8 γ p s 3 / 2 > 0.
r ( b , p s , { u k } ) k = 1 N 1 log ( 1 + u k SNDR k ( b , p s ) ) .
r l b ( p s ; b , p s , { u k } ) = k = 1 N 1 log ( 1 + u k SNDR k ( b , p s ) ) k = 1 N 1 u k SNDR k 2 ( b , p s ) 1 + u k SNDR k ( b , p s ) ( SNDR k 1 ( b , p s ) SNDR k 1 ( b , p s ) ) ,
log ( 1 + 1 / y ) log ( 1 + 1 / x ) 1 x 2 + x ( y x ) .
r l b ( p s ; b , p s , { u k } ) = r ( b , p s , { u k } ) , d d p s r l b ( p s ; b , p s , { u k } ) p s = p s = p s r ( b , p s , { u k } ) p s = p s .
d d p s r l b ( p s ; b , p s , { u k } ) = A q ( p s ) a 3 p s 3 / 2 f d ( p s ; { γ k } ) ,
u k ( p s ) = ( λ ( p s ) 1 SNDR k ( b , p s ) ) + ,
r ( b , p s i , { u k i } ) = ( a ) r l b ( p s i ; b , p s i , { u k i } ) ( b ) r l b ( p s i + 1 ; b , p s i , { u k i } ) ( c ) r ( b , p s i + 1 , { u k i } ) ( d ) r ( b , p s i + 1 , { u k i + 1 } ) ,
h ( 0 ) ( t ; S , R ) = { A PD ( m + 1 ) cos m ϕ 0 cos θ 0 2 π d 0 2 δ ( t d 0 c ) , 0 θ 0 FOV r 0 , θ 0 > FOV r ,
L 1 = A r ( m + 1 ) cos m ϕ 1 cos θ 1 2 π d 1 2 , L 2 = A r cos ϕ 2 cos θ 2 π d 2 2 , , L k + 1 = A PD cos ϕ k + 1 cos θ k + 1 π d k + 1 2
h ( k ) ( t ; S , R ) = { S [ L 1 L 2 L k + 1 Γ ( k ) δ ( t d 1 + d 2 + + d k + 1 c ) ] d A r , k 1 , 0 θ k + 1 FOV r 0 , θ k + 1 > FOV r ,
h ( t ; S , R ) = k = 0 h ( k ) ( t ; S , R ) .
g ( t ) = e 2 π f b t ,

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