PAPR analysis for OFDM visible light communication

Jiaheng Wang; Yang Xu; Xintong Ling; Rong Zhang; Zhi Ding; Chunming Zhao

doi:10.1364/OE.24.027457

1. Introduction

Recently, visible light communication (VLC) has emerged as a promising technology for providing indoor wireless services [1]. Utilizing light-emitting diodes (LEDs) [2], VLC can support illumination and communication simultaneously and offers the benefits of high rates, low cost, safety, security, and no electronic interference. Therefore, VLC has been regarded as an important supplement to, or replacement of, traditional RF communication in indoor environments [3–5].

To fulfill high-speed transmissions, orthogonal frequency-division multiplexing (OFDM), which has been widely used in RF communication systems (e.g., LTE), has also been considered in VLC [6]. However, unlike RF communication that can coherently transmit bipolar and complex-valued signals, VLC generally exploits intensity modulation (IM) and direct detection (DD) to generate real and nonnegative LED-driving signals. Hence, traditional RF OFDM techniques cannot be directly applied to VLC and must be specially modified This reason has led to the rise of various VLC OFDM schemes such as DC-biased optical OFDM (DCO-OFDM) [6], asymmetrically clipped optical OFDM (ACO-OFDM) [7], pulse amplitude modulated discrete multitone (PAM-DMT) [8], and Flip-OFDM [9].

A well known problem of OFDM is its high peak-to-average power ratio (PAPR) [10]. To prevent intermodulation among subcarriers and out-of-band radiation, power amplification at the transmitter should operate in its linear region. A high PAPR requires the power amplifier to possess a wide linear region using a large power back-off, which substantially compromises efficiency. Hence, the PAPR issue has received much attention in RF communication. The PAPR distribution of complex-valued baseband RF-OFDM signals has been studied [11] and various PAPR reduction techniques have also been proposed [10,12–14]. On the other hand, only a few works exist that tackle the PAPR problem in VLC OFDM [13–16]. Thus far, there does not exist a comprehensive study of PAPR distributions of common VLC OFDM schemes.

It is worth noting that existing PAPR analysis for RF-OFDM, e.g., [2], does not apply to VLC OFDM. Time-domain VLC OFDM signals must be real and nonnegative. The nonnegativeness is insured by clipping the signal from below (a DC offset may be added, e.g., in DCO-OFDM). Meanwhile, an LED usually imposes a limit on the maximum magnitude of the input signal, in response to eye safety or device considerations [17,18]. Thus, time-domain VLC OFDM signals may also be clipped from above. Such lower and upper clippings shall be taken into PAPR analysis. In the literature, the cumulative distribution function (CDF) of the unclipped DCO-OFDM signal was investigated in [15]. In [16], the distributions of upper and lower PAPRs of DCO-OFDM, instead of the exact PAPR, were derived and the clipping was replaced by a scaling operation. Therefore, the distributions of the PAPRs of the clipped VLC OFDM signals remain unknown.

In this paper, we investigate the PAPR distributions of four VLC OFDM schemes, including DCO-OFDM, ACO-OFDM, PAM-DMT, and Flip-OFDM. Both lower and upper clippings are considered. We derive analytical expressions of the complementary cumulative distribution functions (CCDFs) of the PAPRs of the clipped VLC OFDM signals. We show that the PAPR CCDF is a piecewise function, which is divided by two points for the asymmetrically clipped DCO-OFDM signal and divided by one point for the symmetrically clipped DCO-OFDM signal as well as the clipped ACO-OFDM, PAM-DMT, and Flip-OFDM signals. The (maximum) dividing point is the PAPR ceiling that defines the maximum value of all possible PAPRs. Such a ceiling exists if and only if both lower and upper clippings are performed. We further investigate the impact of lower and upper clippings on PAPRs and show that both the PAPR CCDF and the PAPR ceiling are monotonic with respect to lower and upper clipping bounds, respectively. These results are particularly useful to determine when lower and upper clippings decrease or increase the PAPR. Finally, the derived analytical PAPR CCDFs and the monotonic properties in terms of clipping bounds of the four VLC OFDM schemes are verified by numerical simulations.

Note that clippings not only affect the PAPR but also introduce distortion to the transmitted signal. In this paper, we focus on analyzing the effect of upper and lower clippings on the PAPR with given clipping bounds, signal power, and DC offset. The interested reader is referred to [17–20] for optimization of the clipping, signal power and DC offset.

2. System models

In the section, we introduce the system models of DCO-OFDM, ACO-OFDM, PAM-DMT, and Flip-OFDM.

2.1. DCO-OFDM

Figure 1 shows the transmitter of the DCO-OFDM system [6]. A serial bit stream is split by a serial-to-parallel (S/P) converter and then mapped to quadrature amplitude modulated (QAM) symbols. Let N be the number of subcarriers and S_k be the symbol on the kth subcarrier. To obtain a real time-domain signal, the frequency-domain Hermitian symmetry is required:

S_{k} = S_{N - k}^{*}, k = 1, \dots, \frac{N}{2} - 1

and the 0th and

\frac{N}{2}

th subcarriers are null, i.e., S₀ = S_N/2 = 0. Hence, the frame structure of DCO-OFDM is as follows

S = [0, S_{1}, S_{2}, \dots, S_{\frac{N}{2} - 1}, 0, S_{\frac{N}{2} - 1}^{*}, \dots, S_{2}^{*}, S_{1}^{*}] .

The time-domain signal is obtained by applying the inverse fast Fourier transform (IFFT) to S_k

s_{n} = \frac{1}{\sqrt{N}} \sum_{k = 0}^{N - 1} S_{k} e^{j \frac{2 π k n}{N}} = \frac{2}{\sqrt{N}} \sum_{k = 1}^{\frac{N}{2} - 1} Re {S_{k} e^{j \frac{2 π k n}{N}}}

for n = 0, . . . , N − 1, where Re{·} denotes the real part of a complex value.

Fig. 1 Transmitter block diagram of the DCO-OFDM VLC System.

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As VLC exploits intensity modulation (IM) to drive an LED, the transmitted signal s_n is required to be not only real but also nonnegative. Thus, a lower clipping and a DC bias are necessary. Moreover, an LED may have a limit on the maximum value of the input signal, e.g., for brightness control or fitting in the linear range of the transfer characteristic, so an upper clipping may also be needed. The clipping process can be expressed as

s_{clip, n} = Clip [s_{n}] = {\begin{array}{l} B_{U}, & s_{n} \geq B_{U} \\ s_{n}, & - B_{L} < s_{n} < B_{U} \\ - B_{L}, & s_{n} \leq - B_{L} \end{array}

where B_U and −B_L represent the upper and lower clipping bounds. The smaller B_U and B_L are, the more the signal is clipped. The clipped signal s_clip,n is fed into the the digital-to-analog convertor (DAC) resulting in the continuous time signal s_clip(t). A DC bias B_L is added onto s_clip(t) to obtain a nonnegative signal s_dc(t) = s_clip(t) + B_L, which is then used to drive an LED.

The receiver uses a photo detector to convert the transmitted signal, which passes through the optical channel, to an electrical signal and then performs direct detection (DD) to restore the transmitted symbols.

2.2. ACO-OFDM

Figure 2 shows the transmitter of the ACO-OFDM system [7]. In ACO-OFDM, only the odd subcarriers carry data symbols and the even subcarriers are set to zero, i.e., S_k = 0 for k = 0, 2, 4, ..., N − 2. At the same time, ACO-OFDM also satisfies the Hermitian symmetry in the frequency domain and thus has the following frame structure

S = [0, S_{1}, 0, \dots, S_{\frac{N}{2} - 1}, 0, S_{\frac{N}{2} - 1}^{*}, \dots, 0, S_{1}^{*}] .

Therefore, the spectral efficiency of ACO-OFDM is a half of that of DCO-OFDM. After the IFFT, the time-domain signal is obtained for n = 0, . . . , N − 1

s_{n} = \frac{1}{\sqrt{N}} \sum_{k = 0}^{N - 1} S_{k} e^{j \frac{2 π k n}{N}} = \frac{2}{\sqrt{N}} \sum_{k = 1}^{\frac{N}{2} - 1} Re {S_{k} e^{j \frac{2 π k n}{N}}} = \frac{2}{\sqrt{N}} \sum_{t = 1}^{N / 4} Re {S_{2 t - 1} e^{j \frac{2 π (2 t - 1) n}{N}}} .

Note that the time-domain signal of ACO-OFDM has following symmetric property:

s_{n + \frac{N}{2}} = \frac{2}{\sqrt{N}} \sum_{t = 1}^{N / 4} Re {S_{2 t - 1} \cdot e^{j \frac{2 π (2 t - 1) (n + \frac{N}{2})}{N}}} = - s_{n}, n = 0, \dots, \frac{N}{2} - 1 .

Therefore, one can transmit the positive part of s_n and clip its negative part without losing any information. Such a clipping process plus the upper bound limit can be expressed as

s_{clip, n} = Clip [s_{n}] = {\begin{array}{l} B_{U}, & s_{n} \geq B_{U} \\ s_{n}, & 0 < s_{n} < B_{U} \\ 0, & s_{n} \leq 0 . \end{array}

Since s_clip,n is nonnegative, there is no need to add a DC bias. The clipped signal is then converted to the continuous signal s_clip(t) for transmitting via an LED.

Fig. 2 Transmitter block diagram of the ACO-OFDM VLC System.

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2.3. PAM-DMT

Figure 3 shows the transmitter of the PAM-DMT system [8]. Different from DCO-OFDM and ACO-OFDM that use complex QAM symbols, PAM-DMT uses PAM symbols as the imaginary parts of transmitted symbols. Specifically, the transmitted symbol on subcarrier k is given by S_k = jA_k for $k = 1, \dots, \frac{N}{2} - 1$ , where A_k is a real PAM symbol. To satisfy the Hermitian symmetry in the frequency domain, the PAM-MDT frame has the following structure

S = [0, j A_{1}, j A_{2}, \dots, j A_{\frac{N}{2} - 1}, 0, - j A_{\frac{N}{2} - 1}, \dots, - j A_{2}, - j A_{1}] .

Thus, PAM-DMT has the same spectral efficiency as ACO-OFDM. After the IFFT, the time-domain signal is obtained

s_{n} = \frac{1}{\sqrt{N}} \sum_{k = 0}^{N - 1} S_{k} e^{j \frac{2 π k n}{N}} = \frac{1}{\sqrt{N}} \sum_{k = 1}^{\frac{N}{2} - 1} [j A_{k} e^{j \frac{2 π k n}{N}} - j A_{k} e^{j \frac{2 π (N - k) n}{N}}] = - \frac{2}{\sqrt{N}} \sum_{k = 1}^{\frac{N}{2} - 1} A_{k} sin \frac{2 π k n}{N}

for n = 0, . . . , N − 1. Note that the time-domain signal of PAM-DMT is also symmetric

s_{N - n} = - \frac{2}{\sqrt{N}} \sum_{k = 1}^{\frac{N}{2} - 1} A_{k} sin \frac{2 π k (N - n)}{N} = - s_{n} n = 1, \dots, \frac{N}{2} - 1

and s₀ = s_N/2 = 0. Therefore, similar to ACO-OFDM, one can transmit the nonnegative part of s_n and clip its negative part without adding a DC bias. The clipped signal s_clip,n is obtained by using the same clipping function in (5) on s_n.

Fig. 3 Transmitter block diagram of the PAM-DMT VLC System.

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2.4. Flip-OFDM

Figure 4 shows the transmitter of the Flip-OFDM system [9]. The transmitting process of Flip-OFDM is the same with that of DCO-OFDM until the IFFT, thus resulting in the same time-domain signal s_n as in (2). Then, s_n is divided into the positive and negative parts as $s_{n} = s_{n}^{+} + s_{n}^{-}$ where

s_{n}^{+} = {\begin{array}{l} s_{n}, & if s_{n} \geq 0 \\ 0, & otherwise \end{array} and s_{n}^{-} = {\begin{array}{l} s_{n}, & if s_{n} < 0 \\ 0, & otherwise . \end{array}

These two parts are transmitted through two frames with the first frame

s_{n}^{1} = s_{n}^{+}

and the second frame

s_{n}^{2} = - s_{n}^{-}

. Hence, the spectral efficiency of Flip-OFDM is equal to that of ACO-OFDM or PAM-DMT, and a half of that of DCO-OFDM.

Fig. 4 Transmitter block diagram of the Flip-OFDM VLC System.

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Let ${\bar{s}}_{n} = [s_{n}^{1}, s_{n + N}^{2}]$ be the expended time-domain signal. Since s̄_n is nonnegative, only an upper clipping is needed

s_{clip, n} = Clip [{\bar{s}}_{n}] = {\begin{array}{l} B_{U}, & s_{n} \geq B_{U} \\ s_{n}, & s_{n} < B_{U} \end{array}

and no DC bias is added. At the receiver, the received time-domain signals of two successive frames, say

y_{n}^{1}

and

y_{n}^{2}

, will be combined as

y_{n} = y_{n}^{1} - y_{n}^{2}

for demodulation.

3. PAPR analysis

In this section, we analyze the PAPRs of the clipped signals of the above four VLC OFDM schemes. The clipped signal s_clip,n will be converted to an analog signal and amplified by a power amplifier to drive an LED (in DCO-OFDM the DC bias is added to the amplified signal). As introduced in Section 1, the implementation of power amplification is significantly influenced by the PAPR of the clipped OFDM signal. According to [10, 11], The PAPR of the clipped signal s_clip,n is defined as

PAPR = \frac{max_{0 \leq n \leq N - 1} {| s_{clip, n} |}^{2}}{E [{| s_{clip, n} |}^{2}]}

which is a random variable and characterized by its CCDF, i.e., the probability P(PAPR > x). In the following, we will derive the PAPR CCDFs of the four VLC OFDM schemes and investigate the impact of clipping.

3.1. PDFs of the clipped signals

To derive the PAPR CCDF, the probability density function (PDF) of the clipped signal is needed. Note that the frequency-domain symbols S_k are in general independently identically distributed (i.i.d.) with zero mean. From the central limit theorem (CLT), for a large subcarrier number N, the time-domain signal s_n obeys a zero-mean Gaussian distribution [11, 15–17, 21], i.e., $s_{n} ~ 𝒩 (0, σ_{s}^{2})$ , with the PDF $p (w) = \frac{1}{\sqrt{2 π} σ_{s}} exp {- \frac{w^{2}}{2 σ_{s}^{2}}}$ , where $σ_{s}^{2}$ is the power (variance) of the time-domain signal. This holds for practical subcarrier numbers (typically N ≥ 128) [6].

Define the normalized lower and upper clipping bounds l ∈ (−∞, 0] and u ∈ [0, +∞) as

l = - \frac{B_{L}}{σ_{s}} and u = \frac{B_{U}}{σ_{s}} .

Denote by

g (x) = \frac{1}{\sqrt{2 π}} e^{- \frac{x^{2}}{2}}

and

Φ (x) = \int_{- \infty}^{x} \frac{1}{\sqrt{2 π}} e^{- \frac{t^{2}}{2}} d t

the PDF and CDF of the standard normal distribution, respectively. Let δ(x) be the Dirac delta function and u(x) be the step function. The clipped DCO-OFDM signal s_clip,n follows a truncated Gaussian distribution [22], whose PDF is given by

f_{dco} (w) = Φ (l) δ (w + B_{L}) + \frac{1}{\sqrt{2 π} σ_{s}} e^{- \frac{w^{2}}{2 σ_{s}^{2}}} [u (w + B_{L}) - u (w - B_{U})] + [1 - Φ (u)] δ (w - B_{U}) .

Note that the clipping process in ACO-OFDM, PAM-DMT, or Flip-OFDM can be regarded as a simplified version in DCO-OFDM by setting B_L = 0. Thus, the PDF of the clipped ACO-OFDM, PAM-DMT, or Flip-OFDM signal is obtained by setting l = 0 in f_dco(w), leading to

f_{aco} (w) = \frac{1}{2} δ (w) + \frac{1}{\sqrt{2 π} σ_{s}} e^{- \frac{w^{2}}{2 σ_{s}^{2}}} [u (w) - u (w - B_{U})] + [1 - Φ (u)] δ (w - B_{U}) .

3.2. Average power of the clipped signal

Given the PDF of the clipped signal, we can compute the average electrical power of s_clip,n, which is the denominator of the PAPR. In particular, the average electrical power of the clipped DCO-OFDM signal is

P_{dco}^{ele} = E [{| s_{clip, n} |}^{2}] = E [s_{clip, n}^{2}] = \int_{- \infty}^{+ \infty} w^{2} f_{dco} (w) d w = σ_{s}^{2} c (l, u)

where c(l, u) is defined as

c (l, u) = (l^{2} - 1) Φ (l) - (u^{2} - 1) Φ (u) + u^{2} + \lg (l) - u g (u) .

The average electrical power of the clipped ACO-OFDM, PAM-DMT, or Flip-OFDM signal is then obtained by setting l = 0 in (9), leading to

P_{aco}^{ele} = σ_{s}^{2} c (0, u)

.

Lemma 1 c(l, u) ∈ [0, 1] is decreasing in l (increasing in B_L) and increasing in u (B_U).

Proof. It can be verified that ∂c(l, u)/∂l = 2lΦ(l) ≤ 0 and ∂c(l, u)/∂u = 2u(1 − Φ(u)) ≥ 0. Therefore, c(l, u) is decreasing in l and increasing in u, and 0 = c(0, 0) ≤ c(l, u) ≤ c(−∞, +∞) = 1. □

It follows from Lemma 1 that $P_{dco}^{ele}$ and $P_{aco}^{ele}$ are increasing in B_U and $P_{dco}^{ele}$ is increasing in B_L. Given the same upper clipping bound B_U, we have $P_{dco}^{ele} \geq P_{aco}^{ele}$ . This is consistent with the intuition that the more the signal is clipped, the less the average electrical power is. On the other hand, it is not straightforward to see the impact of clipping on the PAPR, since the peak power is also reduced by upper and lower clippings.

Moreover, we can also compute the average optical power of the clipped signal, which is given by E[s_clip,n]. In particular, the average optical power of the clipped DCO-OFDM signal is

P_{dco}^{opt} = E [s_{clip, n}] + B_{L} = \int_{- \infty}^{+ \infty} w f_{dco} (w) d w = σ_{s} [l Φ (l) + g (l) - g (u) + u (1 - Φ (u))] + B_{L}

and the average optical power of the clipped ACO-OFDM, PAM-DMT, or Flip-OFDM signal is given by

P_{aco}^{opt} = σ_{s} [\frac{1}{\sqrt{2 π}} - g (u) + u (1 - Φ (u))]

.

3.3. PAPR distribution of DCO-OFDM

Given the PDF and average electrical power of the clipped DCO-OFDM signal, we are able to provide an analytical expression of the PAPR CCDF for DCO-OFDM.

Theorem 1 The PAPR CCDF of the clipped DCO-OFDM signal is

F_{dco} (x) = P (PAPR > x) = {\begin{array}{l} 1 - {[2 Φ (\sqrt{c (l, u) x}) - 1]}^{N}, & 0 \leq x < θ_{min} \\ 1 - {[Φ (\sqrt{c (l, u) x})]}^{N}, & θ_{min} \leq x < θ_{max} \\ 0, & x \geq θ_{max} \end{array}

where θ_min = min{θ_L, θ_U} and θ_max = max{θ_L, θ_U} with

θ_{L} = \frac{l^{2}}{c (l, u)} and θ_{U} = \frac{u^{2}}{c (l, u)} .

Proof. See Appendix A1. □

From Theorem 1, the PAPR CCDF of the clipped DCO-OFDM signal, i.e., F_dco(x), is a piecewise function divided by θ_min and θ_max, which are jointly determined by the normalized lower and upper clipping bounds l and u (or B_L and B_U). If B_L ≤ B_U, then θ_min = θ_L and θ_max = θ_U, otherwise θ_min = θ_U and θ_max = θ_L. Note that F_dco(x) is not continuous at θ_min and θ_max. In particular, F_dco(x) = 0 if x ≥ θ_max, i.e., θ_max is the ceiling of the PAPR. Such a PAPR ceiling exists if only if both lower and upper clippings are performed. If the upper and lower clippings are symmetric, i.e., −l = u(B_L = B_U), then the PAPR CCDF of the clipped DCO-OFDM signal simplifies to

F_{sym} (x) = {\begin{array}{l} 1, - {[2 Φ (\sqrt{c (- u, u) x}) - 1]}^{N}, & 0 \leq x < θ \\ 0, & x \geq θ \end{array}

where θ = u²/c(−u, u). In this case, there is only one dividing point θ, which is the PAPR ceiling.

Proposition 1 θ_L is decreasing in l and u, respectively, and θ_U is increasing in l and u, respectively. For the symmetric clipping (−l = u), θ is increasing in u.

Proof. See Appendix A2. □

Proposition 1 indicates that the dividing points θ_min and θ_max are monotonic in the upper and lower clipping bounds, but depending on the relation between l and u (or B_L and B_U). Specifically, If −l < u (B_L < B_U), then θ_min = θ_L is decreasing in l and u and θ_max = θ_U is increasing in l and u, and vice versa. This reveals that in the asymmetric case, a single-side clipping may not reduce but may enlarge the PAPR ceiling. Indeed, for −l < u, θ_max = θ_U is increasing in l for fixed u, i.e., the more the signal is clipped from below, the larger the PAPR ceiling is. Similarly, for −l > u, θ_max = θ_L is decreasing in u for fixed l, i.e., the more the signal is clipped from above, the larger the PAPR ceiling is. Hence, an effective way to reduce the PAPR ceiling θ_max is to perform both lower and upper clippings. This can be clearly seen from the symmetric clipping −l = u (B_L = B_U), where the PAPR ceiling θ is increasing in u, implying that the more the signal is (symmetrically) clipped, the less the maximum PAPR is.

Proposition 2 For 0 ≤ x < θ_min or θ_min ≤ x < θ_max, F_dco(x) is increasing in l and decreasing in u, respectively. For 0 ≤ x < θ, F_sym(x) is decreasing in u.

Proof. The monotonicity of F_dco(x) in terms of l and u follows directly from Lemma 1. For F_sym(x), it can be verified that c′(−u, u) = 4u(1 − Φ(u)) ≥ 0, so c(−u, u) is increasing in u. Therefore, F_sym(x) is decreasing in u for 0 ≤ x < θ. □

Proposition 2 indicates that the PAPR CCDF of DCO-OFDM is monotonic in the clipping bounds within some range. Specifically, for 0 ≤ x < θ_min or θ_min ≤ x < θ_max, the smaller the clipping bounds B_U and B_L are, i.e., the more the DCO-OFDM signal is clipped, the larger the PAPR could be. However, such a monotonicity does not hold over the whole PAPR range. In fact, from Propositions 1 and 2, we can see a tradeoff in the PAPR caused by clipping. Take the symmetric clipping for example, the more the signal is clipped (i.e., u is smaller), the larger the PAPR could be within [0, θ). On the other hand, a smaller u leads to a smaller θ, i.e., a smaller PAPR ceiling.

From Theorem 1, we can readily obtain the PAPR CCDF of the unclipped signal s_n. Specifically, by setting B_L = B_U = +∞ (l = −∞ and u = +∞), which leads to c(−∞, +∞) = 1 and θ_min = θ_max = +∞, we obtain

F_{unc} (x) = 1 - {[2 Φ (\sqrt{x}) - 1]}^{N}, x \geq 0 .

Proposition 3 Let α = min{θ_max, x₀} and β = max{θ_min, x₀}, where x₀ is the root of the equation

Φ (\sqrt{c (l, u) x}) = 2 Φ (\sqrt{x}) - 1

. Then,

F_{dco} (x) {\begin{array}{l} \geq F_{unc} (x), & x < θ_{min} or β \leq x < θ_{max} \\ \leq F_{unc} (x), & x \geq θ_{max} or θ_{min} \leq x \leq α \end{array}

and

F_{sym} (x) {\begin{array}{l} \geq F_{unc} (x), & 0 \leq x < θ \\ \leq F_{unc} (x), & x \geq θ . \end{array}

Proof. See Appendix A3. □

Proposition 3 indicates that the PAPR CCDF of the clipped DCO-OFDM signal could be higher or lower than that of the unclipped signal within different PAPR ranges. This is actually consistent with Propositions 1 and 2 in the sense that clipping has both positive and negative impacts on the PAPR. The positive aspect is that (double-side) clipping leads to a PAPR ceiling restricting the maximum PAPR, and the negative aspect is that clipping increases the probability of large PAPRs below the ceiling. Note that the nonlinear equation $Φ (\sqrt{c (l, u) x}) = 2 Φ (\sqrt{x}) - 1$ has a unique root, since $Φ (\sqrt{c (l, u) x})$ and $2 Φ (\sqrt{x}) - 1$ are both monotonically increasing in x.

3.4. PAPR distributions of ACO-OFDM and PAM-DMT

Now, we investigate the PAPR distributions of the clipped ACO-OFDM and PAM-DMT signals.

Theorem 2 The PAPR CCDF of the clipped ACO-OFDM signal is

F_{aco} (x) = P (PAPR > x) = {\begin{array}{l} 1 - {[2 Φ (\sqrt{c (0, u) x}) - 1]}^{\frac{N}{2}}, & 0 \leq x < θ_{0, U} \\ 0, & x \geq θ_{0, U} \end{array}

where θ_0,U = u²/c(0, u).

Proof. See Appendix A4. □

It is worth pointing out that F_aco(x) of ACO-OFDM cannot be obtained from F_dco(x) of DCO-OFDM. Indeed, if one sets l = 0 in F_dco(x), he will obtain $F_{dco} (x) = 1 - {[Φ (\sqrt{c (0, u) x})]}^{N}$ for 0 ≤ x < θ_0,U, which is different from $F_{aco} (x) = 1 - {[2 Φ (\sqrt{c (0, u) x}) - 1]}^{\frac{N}{2}}$ for 0 ≤ x < θ_0,U. The reason is that there is a symmetry in the (unclipped) ACO-OFDM signal s_n (see (4)). Therefore, F_aco(x) has to be derived in a different way.

Follow the similar steps, we can show that θ_0,U is increasing in u, i.e., the more the ACO-OFDM signal is clipped from above, the smaller the PAPR ceiling is. On the other hand, F_aco(x) is decreasing in u for 0 ≤ x < θ_0,U, i.e., the more the ACO-OFDM signal is clipped, the more likely a large PAPR appears within [0, θ_0,U). In the case that there is no upper clipping, i.e., u = +∞, we obtain c(0, +∞) = 1/2 and $F_{aco} (x) = 1 - {[2 Φ (\sqrt{x / 2}) - 1]}^{\frac{N}{2}}$ for x ≥ 0.

Theorem 3 The PAPR CCDF of the clipped PAM-DMT signal is

F_{pam} (x) = P (PAPR > x) = {\begin{array}{l} 1 - {[2 Φ (\sqrt{c (0, u) x} - 1]}^{\frac{N}{2} - 1}, & 0 \leq x < θ_{0, U} \\ 0, & x \geq θ_{0, U} \end{array}

where θ_0,U = u²/c(0, u).

The PAPR CCDF of PAM-DMT can be derived in a similar way as that of ACO-OFDM. Indeed, the PAM-DMT signal is clipped in the same manner as ACO-OFDM in (5) and the unclipped PAM-DMT signal also has a symmetry. In particular, the time-domain PAM-DMT signal s_n satisfies s_n = −s_N−n for $n = 1, \dots . \frac{N}{2} - 1$ , which, though a bit different from the symmetry of the ACO-OFDM signal in (4), enables us to use the similar reasoning in deriving the CCDF. The reason why $F_{pam} (x) = 1 - {[2 Φ (\sqrt{c (0, u) x} - 1]}^{\frac{N}{2} - 1}$ instead of $1 - {[2 Φ (\sqrt{c (0, u) x} - 1]}^{\frac{N}{2}}$ for 0 ≤ x < θ_0,U is that for PAM-DMT s₀ and s_N/2 are zero. In practice, for a large number of subcarriers, the PAPR CCDFs of PAM-DMT and ACO-OFDM tend to be equal. Apparently, F_pam(x) shares the same monotonicity in the upper clipping bound u as F_aco(x). In the case that there is no upper clipping, F_pam(x) simplifies to $F_{pam} (x) = 1 - {[2 Φ (\sqrt{x / 2}) - 1]}^{\frac{N}{2} - 1}$ for x ≥ 0.

3.5. PAPR distribution of Flip-OFDM

A complete time-domain Flip-OFDM frame consists of two N-point frames, which correspond to the positive and (flipped) negative parts of the time-domain signal s_n, respectively. Therefore, the PAPR of the clipped Flip-OFDM signal shall be

PAPR = \frac{max_{0 \leq n \leq 2 N - 1} {| s_{clip, n} |}^{2}}{E [{| s_{clip, n} |}^{2}]}

whose CCDF is given in the following result.

Theorem 4 The PAPR CCDF of the clipped Flip-OFDM signal is

F_{flip} (x) = P (PAPR > x) = {\begin{array}{l} 1 - {[2 Φ (\sqrt{c (0, u) x} - 1)]}^{N}, & 0 \leq x < θ_{0, U} \\ 0, & x \geq θ_{0, U} \end{array}

where θ_0,U = u²/c(0, u).

Proof. See Appendix A5. □

Note that the positive and negative parts of the time-domain signal s_n are correlated. This is the reason that the PAPR CCDF of Flip-OFDM has a similar form with those of ACO-OFDM and PAM-DMT, both of which introduce correlation in the time-domain signal as a result of the time-domain symmetry. On the other hand, as a Flip-OFDM frame contains two N-point frames, the exponent in F_flip(x) is N instead of N/2 (as in ACO-OFDM) for 0 ≤ x < θ_0,U. Similar to F_aco(x) and F_pam(x), F_flip(x) is decreasing in u for 0 ≤ x < θ_0,U, while θ_0,U is increasing in u. In the case that there is no upper clipping, F_flip(x) simplifies to $F_{flip} (x) = 1 - {[2 Φ (\sqrt{x / 2}) - 1]}^{N}$ for x ≥ 0. From Theorems 2, 3, and 4, we can obtain the following relation between the PAPR CCDFs of ACO-OFDM, PAM-DMT, and Flip-OFDM

F_{aco} (x) \leq F_{pam} (x) \leq F_{flip} (x)

since

0 \leq 2 Φ (\sqrt{c (0, u) x}) - 1 \leq 1

. In practice, F_aco(x) ≈ F_pam(x) ≤ F_flip(x).

4. Simulation results

In this section, we verify the derived analytical results through numerical simulations. We first show the values of the three important parameters c(l, u), θ_L and θ_U versus the normalized clipping bounds l and u in Fig. 5. Note that c(l, u) is proportional to the average electrical power of the clipped signal (see Section 3.2). From Fig. 5(a), one can observe that c(l, u) is increasing in u and decreasing in l, which is consistent with Lemma 1. θ_L and θ_U are the dividing points of the piecewise PAPR CCDF of the clipped DCO-OFDM signal and PAPR ≤ θ_max = max{θ_L, θ_U}. It can be observed from Fig. 5(b) that θ_L is decreasing in l and u and θ_U is increasing in l and u, which complies with Proposition 1.

Fig. 5 (a) values of c(l, u) and (b) values of θ_L and θ_U for different l and u.

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Figure 6 displays the PAPR CCDFs of the four VLC OFDM schemes with different subcarrier numbers, where DCO-OFDM, ACO-OFDM, and Flip-OFDM exploit 4-QAM modulation and PAM-DMT exploits 4-PAM modulation. For DCO-OFDM, PAM-DMT, and Flip-OFDM, the analytical and simulated PAPR CCDFs match well even if there are only N = 128 subcarriers. In ACO-OFDM the gap between the analytical and simulated CCDFs is a bit larger than that in DCO-OFDM, PAM-DMT, or Flip-OFDM for N = 128. This is because the number of independent frequency-domain symbols is N/2 in DCO-OFDM, PAM-DMT, and Flip-OFDM but N/4 in ACO-OFDM. Therefore, the Gaussian approximation of the time-domain signal is more accurate in DCO-OFDM, PAM-DMT, and Flip-OFDM than in ACO-OFDM given the same subcarrier number. Indeed, when the subcarrier number is equal to or larger than 256, the analytical and simulated ACO-OFDM PAPR CCDFs also match well. Note that analytical and simulated CCDFs have exactly the same PAPR ceiling, which indicates the accuracy of our analytical results too.

Fig. 6 PAPR CCDFs of the four VLC OFDM schemes with different subcarrier numbers.

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In Fig. 7, we plot the simulated and analytical PAPR CCDFs of the clipped and unclipped DCO-OFDM signals with N = 1024 subcarriers. For the asymmetric clipping, the PAPR CCDF has two dividing points, i.e., θ_min and θ_max. For l = −5 and u = 0, . . . , 4, θ_min = θ_U is increasing in u and θ_max = θ_L is decreasing in u. For u = 5 and l = 0, . . . , −4, θ_min = θ_L is decreasing in l and θ_max = θ_U is increasing in l. It is consistent with Proposition 1 and also demonstrates that single-side clippings may increase the PAPR ceiling, i.e., the more the signal is clipped from one side (above or below), the larger θ_max becomes. For the symmetric clipping, the PAPR CCDF has only one dividing point θ, which is increasing in u. In this case, the more the signal is clipped, the smaller the PAPR ceiling θ is. The PAPR CCDF of the clipped DCO-OFDM signal could be smaller or larger than that of the unclipped signal (l = −∞, u = ∞), depending on the specific PAPR regions given in Proposition 3.

Fig. 7 DCO-OFDM PAPR CCDF versus PAPR with N = 1024.

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Figure 8 displays the PAPR CCDF of DCO-OFDM versus the clipping bounds l and u, respectively. From Figs. 8(b) and 8(a), one can observe that the CCDF is increasing in l and decreasing in u within some region, which is consistent with Proposition 2. Note that such a monotonicity only holds for the PAPR within [0, θ_min) or [θ_min, θ_max). When l or u changes, the dividing points θ_min and θ_max also change. Hence, a given PAPR may move out the region [0, θ_min) or [θ_min, θ_max), which leads the discontinuity in Figs. 8(a) and 8(b).

Fig. 8 DCO-OFDM PAPR CCDF versus l and u with N = 1024. In (a) u = 5 and in (b) l = −5.

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In Fig 9, we plot the simulated and analytical PAPR CCDFs of the clipped ACO-OFDM, PAM-DMT, and Filp-OFDM signals with N = 1024 subcarriers. Given the same upper clipping bound u, the clipped ACO-OFDM, PAM-DMT, and Filp-OFDM signals have the same PAPR ceiling θ_0,U, which is increasing in u, i.e., the more the signal is clipped above, the smaller the maximum PAPR is. The PAPR CCDFs of ACO-OFDM and PAM-DMT are nearly identical and smaller than that of Flip-OFDM. This is consistent with the relation in (19).

Fig. 9 ACO-OFDM, PAM-DMT, Flip-OFDM PAPR CCDFs versus PAPR with N = 1024.

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In Fig. 10, we compare the PAPR CCDFs of the clipped DCO-OFDM, ACO-OFDM, PAM-DMT, and Flip-OFDM signals. Given the same upper clipping bound u, the PAPR CCDF of the half-clipped (l = 0) DCO-OFDM signal is quite similar to that of the clipped ACO-OFDM or PAM-DMT signal (therefore their PAPR CCDF curves overlap in Fig. 10), while Flip-OFDM has the largest PAPR CCDF. However, the DCO-OFDM signal is very unlikely to be half clipped from below, which will cause a great deal of information loss. Therefore, in practice, we have l < 0 for DCO-OFDM and in this case, from Figs. 10(a)–10(c), DCO-OFDM has a smaller PAPR than ACO-OFDM, PAM-DMT, and Flip-OFDM. From Figs. 10(c) and 10(d), one can observe that when the normalized clipping bound is equal to 5, the effect of upper or lower clipping tends to saturation.

Fig. 10 Comparison of the PAPR CCDFs of the four VLC OFDM schemes.

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5. Conclusion

We investigated PAPR distributions of the clipped DCO-OFDM, ACO-OFDM, PAM-DMT, and Flip-OFDM signals and took into account both lower and upper clippings. The PAPR CCDFs of the four VLC OFDM schemes were analytically characterized and shown to be piecewise functions. We further investigated the monotonic properties of the PAPR CCDF and the PAPR ceiling with respect to the lower and upper clipping bounds, which lead to useful insights for VLC OFDM systems. The derived analytical results were verified by numerical simulations.

Appendix

A1. Proof of theorem 1

From (9) we know $E [{| s_{clip, n} |}^{2}] = σ_{s}^{2} c (l, u)$ . The CDF of the PAPR is

P (PAPR \leq x) = P (max_{0 \leq n \leq N - 1} {| s_{clip, n} |}^{2} \leq σ_{s}^{2} c (l, u) x) = P ({| s_{clip, n} |}^{2} \leq σ_{s}^{2} c (l, u) x, n = 0, \dots, N - 1)

Although the Hermitian symmetry introduces correlation among s_n, for a large N the correlation tends to be zero [21], i.e., s_n (s_clip,n) is independent for n = 0, . . . , N − 1. Hence, we have

P (PAPR \leq x) = {[P ({| s_{clip, n} |}^{2} \leq σ_{s}^{2} c (l, u) x)]}^{N} .

Consider the following four situations: 1) If

σ_{s} \sqrt{c (l, u) x} < min {B_{L}, B_{U}}

, then

\begin{array}{l} P ({| s_{clip, n} |}^{2} \leq σ_{s}^{2} c (l, u) x) & = P (- σ_{s} \sqrt{c (l, u) x} \leq s_{clip, n} \leq σ_{s} \sqrt{c (l, u) x}) \\ = \int_{- σ_{s} \sqrt{c (l, u) x}}^{σ_{s} \sqrt{c (l, u) x}} f_{dco} (w) d w = 2 Φ (\sqrt{c (l, u) x}) - 1 . \end{array}

2) If

σ_{s} \sqrt{c (l, u) x} \geq max {B_{L}, B_{U}}

, then

P ({| s_{clip, n} |}^{2} \leq σ_{s}^{2} c (l, u) x) = P (- B_{L} \leq s_{clip, n} \leq B_{U}) = \int_{- B_{L}}^{B_{U}} f_{dco} (w) d w = 1 .

3) If

B_{L} = min {B_{L}, B_{U}} \leq σ_{s} \sqrt{c (l, u) x} < max {B_{L}, B_{U}} = B_{U}

, then

\begin{array}{l} P ({| s_{clip, n} |}^{2} \leq σ_{s}^{2} c (l, u) x) & = P (- B_{L} \leq s_{clip, n} \leq σ_{s} \sqrt{c (l, u) x}) \\ = \int_{- B_{L}}^{σ_{s} \sqrt{c (l, u) x}} f_{dco} (w) d w = Φ (\sqrt{c (l, u) x}) . \end{array}

4) If

B_{U} = min {B_{L}, B_{U}} \leq σ_{s} \sqrt{c (l, u) x} < max {B_{L}, B_{U}} = B_{L}

, then

\begin{array}{l} P ({| s_{clip, n} |}^{2} \leq σ_{s}^{2} c (l, u) x) & = P (- σ_{s} \sqrt{c (l, u) x} \leq s_{clip, n} \leq B_{U}) \\ = \int_{- σ_{s} \sqrt{c (l, u) x}}^{B_{U}} f_{dco} (w) d w = Φ (\sqrt{c (l, u) x}) . \end{array}

Therefore, the CDF of the PAPR is given by

P (PAPR \leq x) = {\begin{array}{l} {[2 Φ (\sqrt{c (l, u) x} - 1)]}^{N}, & x < θ_{min} \\ {[Φ (\sqrt{c (l, u) x})]}^{N}, & θ_{min} \leq x < θ_{max} \\ 1, & x \geq θ_{max} . \end{array}

The CCDF in (11) is 1 − P(PAPR ≤ x).

A2. Proof of proposition 1

It is easily seen that θ_L is decreasing in u, since c(l, u) is increasing in u. For l, we have

\frac{\partial θ_{L}}{\partial l} = \frac{2 l c (l, u) - l^{2} 2 l Φ (l)}{c^{2} (l, u)} = \frac{2 l (c (l, u) - l^{2} Φ (l))}{c^{2} (l, u)} = \frac{2 l ψ_{L} (l, u)}{c^{2} (l, u)}

where ψ_L (l, u) = c(l, u) − l²Φ(l), which is increasing u. It follows that

\frac{\partial ψ_{L} (l, u)}{\partial l} = 2 l Φ (l) - 2 l Φ (l) - l^{2} g (l) = - l^{2} g (l) \leq 0,

implying that ψ_L (l, u) is decreasing in l. So we have ψ_L (l, u) ≥ ψ_L (0, 0) = c(0, 0) = 0 and ∂θ_L/∂l ≤ 0. Therefore, θ_L is decreasing in l.

Similarly, θ_U is increasing in l, since c(l, u) is decreasing in l. For u, we have

\frac{\partial θ_{U}}{\partial u} = \frac{2 u c (l, u) - u^{2} 2 u (1 - Φ (u))}{c^{2} (l, u)} = \frac{2 u (c (l, u) - u^{2} + u^{2} Φ (u))}{c^{2} (l, u)} = \frac{2 u ψ_{U} (l, u)}{c^{2} (l, u)}

where ψ_U (l, u) = c(l, u) − u² + u²Φ(u), which is decreasing in l. It follows that

\frac{\partial ψ_{U} (l, u)}{\partial u} = 2 u (1 - Φ (u)) - 2 u + 2 u Φ (u) + u^{2} g (u) = u^{2} g (u) \geq 0,

implying that ψ_U (l, u) is increasing in u. So we have ψ_U (l, u) ≥ ψ_U (0, 0) = c(0, 0) = 0 and ∂θ_U/∂u ≥ 0. Therefore, θ_U is increasing in u.

Now, consider the symmetric case where −l = u and θ = u²/c(−u, u). It can be verified that dc(−u, u)/du = 4u(1 − Φ(u)). We have

\frac{d θ}{d u} = \frac{2 u c (- u, u) - u^{2} 4 u (1 - Φ (u))}{c^{2} (- u, u)} = \frac{2 u (2 Φ (u) - 2 u g (u) - 1)}{c^{2} (- u, u)} = \frac{2 u ψ (u)}{c^{2} (- u, u)}

where ψ(u) = 2Φ(u) − 2ug(u) − 1. It follows that ψ′(u) = 2u²g(u) ≥ 0, implying that ψ(u) is increasing in u and ψ(u) ≥ ψ(0) = 0. Therefore, dθ/du ≥ 0 and θ is increasing in u.

A3. Proof of proposition 3

If x ≥ θ_max, then G_dco(x) = 0 ≤ G_unc(x). If x < θ_min, then $G_{dco} (x) = 1 - {[2 Φ (\sqrt{c (l, u)} x) - 1]}^{N}$ . In this case, since c(l, u) ≤ c(−∞, +∞) = 1, we obtain $Φ (\sqrt{c (l, u) x}) \leq Φ (\sqrt{x})$ and thus G_dco(x) ≥ G_unc(x). If θ_min ≤ x < θ_max, then $G_{dco} (x) = 1 - {[Φ (\sqrt{c (l, u) x})]}^{N}$ . In this case, G_dco(x) ≤ G_unc(x) if $Φ (\sqrt{c (l, u) x}) \geq 2 Φ (\sqrt{x}) - 1$ , otherwise G_dco(x) ≥ G_unc(x).

Consider the symmetric clipping. If x ≥ θ, then G_sym(x) = 0 ≤ G_unc(x). If 0 ≤ x < θ, then $G_{sym} (x) = 1 - {[2 Φ (\sqrt{c (- u, u) x}) - 1]}^{N}$ . In this case, since c(−u, u) ≤ 1, we obtain $Φ (\sqrt{c (- u, u) x}) \leq Φ (\sqrt{x})$ and thus G_sym(x) ≥ G_unc(x).

A4. Proof of theorem 2

From Section 3.2, we know $E [{| s_{clip, n} |}^{2}] = σ_{s}^{2} c (0, u)$ . The CDF of the ACO-OFDM PAPR is

P (PAPR \leq x) = P (max_{0 \leq n \leq N - 1} {| s_{clip, n} |}^{2} \leq σ_{s}^{2} c (0, u) x) = P (s_{clip, n} \leq σ_{s} \sqrt{c (0, u) x}, n = 0, \dots, N - 1)

where we have used the fact that s_clip,n is nonnegative. It follows from (4) that

s_{n + \frac{N}{2}} = - s_{n}

for

n = 0, \dots, \frac{N}{2} - 1

, i.e., s_n and

s_{n + \frac{N}{2}}

are correlated, so are s_clip,n and

s_{clip, n + \frac{N}{2}}

. On the other hand, for a large N, s_n (s_clip,n) is independent for

n = 0, \dots, \frac{N}{2} - 1

. Therefore, we have

P (PAPR \leq x) = \prod_{n = 0}^{\frac{N}{2} - 1} P (s_{clip, n} \leq σ_{s} \sqrt{c (0, u) x}, s_{clip, n + \frac{N}{2}} \leq σ_{s} \sqrt{c (0, u) x}) .

It follows that

\begin{array}{l} P (s_{clip, n} \leq σ_{s} \sqrt{c (0, u) x}, s_{clip, n + \frac{N}{2}} \leq σ_{s} \sqrt{c (0, u) x}) \\ = P (s_{clip, n} \leq σ_{s} \sqrt{c (0, u) x}, s_{clip, n + \frac{N}{2}} \leq σ_{s} \sqrt{c (0, u) x}, s_{n} \geq 0) \\ + P (s_{clip, n} \leq σ_{s} \sqrt{c (0, u) x}, s_{clip, n + \frac{N}{2}} \leq σ_{s} \sqrt{c (0, u) x}, s_{n} < 0) \\ \underset{=}{(a)} P (s_{clip, n} \leq σ_{s} \sqrt{c (0, u) x}, s_{n} \geq 0) + P (s_{clip, n + \frac{N}{2}} \leq σ_{s} \sqrt{c (0, u) x}, s_{n} < 0) \\ \underset{=}{(b)} 2 P (s_{clip, n} \leq σ_{s} \sqrt{c (0, u) x}, s_{n} \geq 0) \end{array}

where (a) is because if s_n ≥ 0, then

s_{clip, n + \frac{N}{2}} = 0

and if s_n < 0, then s_clip,n = 0, and (b) is because s_clip,n and

s_{clip, n + \frac{N}{2}}

have the same distribution and P(s_n ≥ 0) = P(s_n < 0) = 0.5. If

σ_{s} \sqrt{c (0, u) x} \geq B_{U}

, then

P (s_{clip, n} \leq σ_{s} \sqrt{c (0, u) x}, s_{n} \geq 0) = P (s_{n} \geq 0) = 0.5

. If

σ_{s} \sqrt{c (0, u) x} < B_{U}

, then

\begin{array}{l} P (s_{clip, n} \leq σ_{s} \sqrt{c (0, u) x}, s_{n} \geq 0) & = P (0 \leq s_{n} \leq σ_{s} \sqrt{c (0, u) x}) = \int_{0}^{σ_{s} \sqrt{c (0, u) x}} \frac{1}{\sqrt{2 π} σ_{s}} e^{\frac{w^{2}}{2 σ_{s}^{2}}} d w \\ = Φ (\sqrt{c (0, u) x}) - 0.5 . \end{array}

Consequently, the CDF of the PAPR is

P (PAPR \leq x) = {\begin{array}{l} {[2 Φ (\sqrt{c (0, u) x}) - 1]}^{\frac{N}{2}}, & 0 \leq x < θ_{0, U} \\ 1, & x \geq θ_{0, U} \end{array}

which leads to the CCDF in (16).

A5. Proof of theorem 4

It follows from Section 3.2 that $E [s_{flip, n}^{2}] = σ_{s}^{2} c (0, u)$ . The CDF of the PAPR is

\begin{array}{l} P (PAPR \leq x) & = P (s_{clip, n} \leq σ_{s} \sqrt{c (0, u) x}, n = 0, \dots, 2 N - 1) \\ = \prod_{n = 0}^{N - 1} P (s_{clip, n} \leq σ_{s} \sqrt{c (0, u) x}, s_{clip, n + N} \leq σ_{s} \sqrt{c (0, u) x}) \end{array}

where we have used the facts that s_clip,n is nonnegative and for a large N, s_n(s_clip,n) is independent for n = 0, . . . , N − 1, but s_clip,n and s_clip,n+N are correlated.

\begin{array}{l} P (s_{clip, n} \leq σ_{s} \sqrt{c (0, u) x}, s_{clip, n + N} \leq σ_{s} \sqrt{c (0, u) x}) \\ = P (s_{clip, n} \leq σ_{s} \sqrt{c (0, u) x}, s_{clip, n + N} \leq σ_{s} \sqrt{c (0, u) x}, s_{n} \geq 0) \\ + P (s_{clip, n} \leq σ_{s} \sqrt{c (0, u) x}, s_{clip, n + N} \leq σ_{s} \sqrt{c (0, u) x}, s_{n} < 0) \\ \underset{=}{(a)} P (s_{clip, n} \leq σ_{s} \sqrt{c (0, u) x}, s_{n} \geq 0) + P (s_{clip, n + N} \leq σ_{s} \sqrt{c (0, u) x}, s_{n} < 0) \\ \underset{=}{(b)} 2 P (s_{clip, n} \leq σ_{s} \sqrt{c (0, u) x}, s_{n} \geq 0) \end{array}

where (a) is because if s_n ≥ 0, then s_clip,n+N = 0 and if s_n < 0, then s_clip,n = 0, and (b) is because s_clip,n and s_clip,n+N have the same distribution and P(s_n ≥ 0) = P(s_n < 0) = 0.5. Then, following the similar steps in Appendix A4, we can obtain the CCDF in (18).

Funding

973 Program of China (2013CB329204); National Natural Science Foundation of China (61571107 and 61521061); the Natural Science Foundation of Jiangsu Province (BK20160069); The Alexander von Humboldt Foundation; Fundamental Research Funds for the Central Universities; EPSRC (EP/N004558/1 and EP/N023862/1).

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PAPR analysis for OFDM visible light communication

Abstract

1. Introduction

2. System models

2.1. DCO-OFDM

2.2. ACO-OFDM

2.3. PAM-DMT

2.4. Flip-OFDM

3. PAPR analysis

3.1. PDFs of the clipped signals

3.2. Average power of the clipped signal

3.3. PAPR distribution of DCO-OFDM

3.4. PAPR distributions of ACO-OFDM and PAM-DMT

3.5. PAPR distribution of Flip-OFDM

4. Simulation results

5. Conclusion

Appendix

A1. Proof of theorem 1

A2. Proof of proposition 1

A3. Proof of proposition 3

A4. Proof of theorem 2

A5. Proof of theorem 4

Funding

References and links

Cited By

Figures (10)

Equations (50)

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