Clipping Noise in Visible Light Communication Systems with OFDM and PAPR Reduction

Abstract

This paper presents an analytical study of signal clipping that leads to the noise/distortion in the waveform of DC-biased optical orthogonal frequency division multiplexing (DCO-OFDM)-based visible light communication (VLC) systems. The pilot-assisted (PA) technique is used to reduce the high peak-to-average power ratio (PAPR) of the time-domain waveform of the DCO-OFDM system. The bit error rate (BER) performance of the PA DCO-OFDM system is investigated analytically at three different clipping levels as well as without any clipping. The analytical BER performance is verified through simulation and then compared to that of the conventional DCO-OFDM without PAPR reduction at the selected clipping levels. The PA DCO-OFDM system shows improved BER performance at all three clipping levels.

1. Introduction

The high interest in the visible light communication (VLC) continues due to its huge and unlicensed optical spectrum which can be used for high rate wireless data transmission. VLC systems use low-cost off-the-shelf front-end components, such as light-emitting diodes (LEDs) and photodetectors (PDs). The VLC technology is an attractive complementary solution to the limited availability of the radio frequency (RF) spectrum. This makes the VLC system a potential candidate for many applications such as those related to the Internet of Things (IoT), underwater communications, and indoor positioning. Furthermore, the VLC system is suitable for use in homes, offices, and places such as hospitals and areas where RF wireless and wired communications are not suitable. In addition, by reusing the existing lighting infrastructures for connectivity, VLC can enable significant cost and energy saving [1,2,3,4].

Direct current-biased optical orthogonal frequency division multiplexing (DCO-OFDM) is widely studied modulation candidate for VLC implementation. This is due to its advantages such as intersymbol interference (ISI) mitigation, resilience to multipath propagation, robustness of the system against frequency selectivity of the channel and simplified equalisation process [5,6,7]. In addition, the light source such as LEDs require direct current (DC) bias to turn on the light sources which already exists in DCO-OFDM for the purpose of converting the bipolar waveform to unipolar one. However, DCO-OFDM suffers from high peak-to-average power ratio (PAPR) due to the coherently added up subcarriers in the time domain [5]. The high PAPR of the orthogonal frequency division multiplexing (OFDM) waveform and the limited operating dynamic range of the optical light source result in clipping noise and nonlinear distortion of the transmitted signal [8].

Thus, implementing a PAPR reduction technique will help minimise the clipping noise and nonlinear distortion. Various solutions for PAPR reduction have been proposed in the literature. These include signal clipping methods in [5,9], and the classical selected mapping (SLM) algorithm in [5,9,10]. Other methods based on the SLM scheme are reported in [11,12]. The tone injection (TI) scheme is another PAPR reduction method studied in [9], and the tone reservation (TR) technique in [13]. Pilot-assisted (PA) technique was proposed in [5], and it is considered an effective solution for PAPR reduction in optical OFDM-based systems. The PA approach is applied in this work to reduce the high PAPR peaks of the transmitted waveform in PA DCO-OFDM [5]. The PA scheme rotates the phase of the transmitted OFDM data frame using a pilot sequence that is randomly generated to avoid adding the subcarriers coherently as much as possible. The used pilot sequence phase is selected based on the SLM algorithm and recovered at the receiver side by maximum likelihood (ML) algorithm [14].

The PAPR reduction capabilities of PA technique is studied through simulation in [5], then compared to most widely used PAPR reduction techniques of clipping and classical SLM and the PAPR reduction impact on the system’s bit error rate (BER) performance is investigated. In [6], the PA technique is implemented to reduce the peak values of the pulse-amplitude modulated discrete multitone (PAM-DMT)-based VLC system and the system’s error performance is experimentally investigated using a blue LED. In [15], the analytical framework of the PAPR distribution of a PA DCO-OFDM and the theoretical analysis of the required DC bias to convert the bipolar PA DCO-OFDM signal to unipolar are presented. In addition, the derivation of the PAPR distribution of PA asymmetrically clipped optical orthogonal frequency division multiplexing (ACO-OFDM) signal is obtained and the average optical power reduction gain of the PA optical OFDM is investigated. In contrast, this work presents an analytical study of the impact of reducing the high PAPR values on signal clipping that leads to the noise/distortion of the time-domain waveform of DCO-OFDM-based VLC system.

In this work, the DCO-OFDM modulation technique with the PAPR reduction is considered (PA DCO-OFDM) and the effect of double-sided clipping on its time-domain waveform is investigated. The attenuation factor and the variance of the clipping noise at the received waveform are determined in closed-form, which are included in the derivation of the signal-to-noise ratio (SNR) of the system. The bit error rate (BER) performance of the proposed PA DCO-OFDM system is analytically investigated at three different clipping levels as well as without any clipping. The analytical BER performance is verified through simulation and then compared to that of the conventional DCO-OFDM without a PAPR reduction at the selected clipping levels. Specific contributions of this manuscript are as follows:

• The analytical investigation of a PAPR reduction in PA DCO-OFDM-based VLC system using the PA technique in the presence of clipping noise.
• The PA technique is applied to reduce clipping/nonlinear distortion and improve the error performance of the system.
• The BER performance of the PA DCO-OFDM is analytically investigated at different clipping levels and compared to the performance of conventional DCO-OFDM without PAPR reduction.
• The analytical comparison is verified through simulation at the selected clipping levels for the compared systems.

The PA DCO-OFDM-based VLC system has achieved a better BER performance compared to that of conventional DCO-OFDM without PAPR reduction.

The rest of this manuscript is organised as follows: The VLC system is presented in Section 2, while the PAPR reduction procedure is presented in Section 3. The PA DCO-OFDM clipping noise analysis is described in Section 4, and the results and discussions are in Section 5. The paper is concluded in Section 6.

2. System Description

The system model considered for this work is illustrated in Figure 1. First, a random bit stream is generated and encoded into complex-valued quadrature amplitude modulation (QAM) symbols $X\left[m\right]$ for $m=1,2,\dots ,\frac{N}{2}-1$ is the active subcarriers index in the OFDM frame, where N is the OFDM frame size. The number of bits, b, that are mapped on each M-QAM symbol is given by $b={log}_2\left(M\right)$. The conventional OFDM waveform is complex and bipolar. However, the transmission mechanism in the optical wireless communication system is intensity modulation with direct detection (IM/DD), which requires a real and positive waveform. This requirement is met by imposing Hermitian symmetry on the OFDM frame in the frequency domain, $X\left[m\right]=X^{*}[N-m]$. The OFDM frame in the frequency domain is given by [16]:

$$X\left[k\right]=\left[0,X_1,\dots ,X_{\frac{N}{2}-1},0,X_{\frac{N}{2}-1}^{*},\dots ,X_1^{*}\right]$$

(1)

where k is the subcarrier index with size N and $X\left[0\right]=X[N/2]=0$. The utilisation factor for the OFDM frame’s double-sided bandwidth B is defined by $G_{\mathrm{B}}$ which is given as: $G_{\mathrm{B}}=(N-2)/N$. The total number of enabled subcarriers on the OFDM frame is $G_{\mathrm{B}}\times N$. The OFDM frame activates all subcarriers which can enable the different modulation format, M, across the OFDM frame. The average electrical power of the M-QAM symbol, $P_{\mathrm{s}\left(\mathrm{elec}\right)}=P_{\mathrm{b}\left(\mathrm{elec}\right)}\times {log}_2\left(M\right)$, where $P_{\mathrm{b}\left(\mathrm{elec}\right)}$ is the average electrical power per bit of the M-QAM symbol. The high PAPR peaks of the time-domain signal are reduced using PA technique by rotating the phase of $X\left[k\right]$ prior to the inverse fast Fourier transform (IFFT) process, as detailed in Section 3. The $X\left[k\right]$ symbols are then modulated into orthogonal subcarriers by applying an IFFT of size N to obtain a discrete time-domain signal $x\left[n\right]$ as follows

$$x\left[n\right]=\frac{1}{\sqrt{N}}\sum _{k=0}^{N-1}X\left[k\right]e^{j2\pi \frac{kn}{N}};0\le n\le N-1$$

(2)

where $j=\sqrt{(-1)}$. In general, a cyclic prefix (CP) is added to the OFDM-based systems to mitigate inter-carrier interference (ICI) and ISI. This is performed by copying the last few samples of the OFDM frame to the beginning of that frame. The CP length, $N_{\mathrm{CP}}$, must be greater than the channel delay spread to mitigate ISI, and hence, the frame length becomes ($N+N_{\mathrm{CP}}$) [17]. For simplicity, the CP is omitted in this work, as it has negligible effects on the electrical SNR, spectral efficiency, and electrical PAPR values of the system [5].

The discrete time-domain signal $x\left[n\right]$ is approximately Gaussian distributed with zero mean and a variance of ${\sigma }_x^2$ according to the central limit theorem (CLT) [18]. The average electrical power of the time domain signal $x\left[n\right]$ is proportional to its variance, ${\sigma }_x^2$ [19]. Therefore, to achieve a specific SNR, $x\left[n\right]$ should be scaled properly in relation with ${\sigma }_x^2$ [8]. Hence, the signal is scaled by a factor, $\alpha$, to produce a specific ${\sigma }_{x_{\mathrm{scaled}}}^2$ of the scaled signal $x_{\mathrm{scaled}}\left[n\right]$. The variance, ${\sigma }_{x_{\mathrm{scaled}}}^2$, of the scaled signal is given as follows

$${\sigma }_{x_{\mathrm{scaled}}}^2={\alpha }^2\times {\sigma }_x^2$$

(3)

Light sources such as LEDs have a limited dynamic range between [$i_{\min }$, $i_{\max }$], corresponding to minimum and maximum current points, respectively. To efficiently utilise that dynamic range, the structure-specific signal pre-clipping has to be performed within the range $[k_{\mathrm{b}},k_{\mathrm{t}}]$, which denotes the predefined bottom and top clipping levels, respectively, as shown in Figure 2. Then, the pre-clipped PA DCO-OFDM signal is biased to fit the linear dynamic range of the LED by a bias current, $i_{\mathrm{bias}}$. The biasing process is employed to facilitate the minimal bottom and top levels signal clipping by placing the signal mean in the middle of the LED’s linear dynamic region [8]. The pre-clipping levels can be defined as follows

$$\begin{array}{cc}\hfill k_{\mathrm{b}}=i_{\min }-i_{\mathrm{bias}},& k_{\mathrm{t}}=i_{\max }-i_{\mathrm{bias}}\hfill \end{array}$$

(4)

In the zero signal clipping scenario, the signal clipping is defined as: $k_{\mathrm{b}}=-\infty$, and $k_{\mathrm{t}}=+\infty$. The clipping levels of the PA DCO-OFDM waveform can be negative and/or positive, as long as the two clipping levels are not equal, $k_{\mathrm{b}}<k_{\mathrm{t}}$. In insufficient biasing scenarios, the signal is pre-clipped at the bottom level, $k_{\mathrm{b}}$. Furthermore, the signal is pre-clipped at the top level, $k_{\mathrm{t}}$. Therefore, the clipped signal $x_{\mathrm{clip}}\left[n\right]$ can be expressed as follows:

$$x_{\mathrm{clip}}\left[n\right]=\left\{\begin{array}{cc}{k}_{\mathrm{t}}\hfill & \mathrm{if}{x}_{\mathrm{scaled}}\left[n\right]\ge k_{\mathrm{t}},\hfill \\ x_{\mathrm{scaled}}\left[n\right]\hfill & \mathrm{if}{k}_{\mathrm{b}}<x_{\mathrm{scaled}}\left[n\right]<k_{\mathrm{t}},\hfill \\ k_{\mathrm{b}}\hfill & \mathrm{if}{x}_{\mathrm{scaled}}\left[n\right]\le k_{\mathrm{b}}.\hfill \end{array}\right.$$

(5)

When an OFDM time-domain signal is clipped, it changes its mean, and as a result, its average optical power. To convert the electrical signal into an optical one and achieve maximum signal power, the linear dynamic range $P_{\min }$ and $P_{\max }$ should be mapped by the clipping levels, $k_{\mathrm{b}}$ and $k_{\mathrm{t}}$, as depicted in Figure 2. Hence, to form such a mapping, an input biasing power $P_{\mathrm{bias}}=\beta \times i_{\mathrm{bias}}$, is required and can be expressed as

$$\begin{array}{cc}\hfill P_{\min }=\beta (k_{\mathrm{b}}+i_{\mathrm{bias}}),& P_{\max }=\beta (k_{\mathrm{t}}+i_{\mathrm{bias}})\hfill \end{array}$$

(6)

where $\beta$ is the electrical-to-optical conversion coefficient.

The clipped continuous time-domain waveform, $x\left(t\right)$, is obtained from $x_{\mathrm{clip}}\left[n\right]$ following the digital-to-analogue conversion (DAC) process to modulate the intensity of the optical source. In general, the entire forward voltage range across the LED is supported by a sufficient constant bias source which converts the bipolar time-domain baseband waveform to a unipolar signal. The bias current is combined with the data waveform current, producing the total forward current through the LED. Since the radiated optical power by the LED is proportional to the forward current, the data waveform, the bias current $i_{\mathrm{bias}}$, and the system constraints imposed by the front-end devices are described in terms of optical power [8]. Therefore, the average transmitted optical power from the LED is given as

$$P_{\mathrm{opt}}=\beta \times \mathrm{E}\left[\mathrm{x}\left(\mathrm{t}\right)\right]+{\mathrm{P}}_{\mathrm{bias}}$$

(7)

The PA DCO-OFDM frame is assumed to be transmitted within the 3 dB bandwidth of the LED; hence, the optical channel is assumed flat over the entire transmitted OFDM frame. The signal is unlikely to be clipped at the receiver side. For example, the photodetector (PD) has a linear dynamic range much higher than that of the transmitter, which needs a very high amount of radiated power in order to drive the PD to its saturation region. At the receiver, a PD converts the received optical radiation into a current signal prior to the trans-impedance amplifier (TIA) which converts the current signal into the corresponding voltage. The voltage signal is then amplified to an appropriate level to obtain $y\left(t\right)$ prior to the analogue to digital conversion (ADC) process. The received time-domain waveform $y\left(t\right)$ is given by:

$$y\left(t\right)=h\left(t\right)⊛x\left(t\right)+w_{\mathrm{rx}}\left(t\right)$$

(8)

where $h\left(t\right)$ is the channel impulse response, $w_{\mathrm{rx}}\left(t\right)$ is the received white Gaussian noise with variance ${\sigma }_{w_{\mathrm{rx}}}^2$, and ⊛ denotes a convolution operation. The received signal $y\left(t\right)$ is then converted to a discrete time-domain signal $y\left[n\right]$ by the ADC block followed by serial-to-parallel (S/P) conversion. The received signal is distorted by zero-mean real-valued bipolar additive white Gaussian noise (AWGN) $w_{\mathrm{rx}}\left(t\right)$ which consists of the receiver’s shot and thermal noises. The fast Fourier transformation (FFT) is applied on $y\left[n\right]$ which transforms the signal back to the frequency domain, which is given by:

$$Y\left[k\right]=\frac{1}{\sqrt{N}}\sum _{n=0}^{N-1}y\left[n\right]e^{-j2\pi \frac{kn}{N}};0\le k\le N-1$$

(9)

The obtained frequency-domain symbol $Y\left[k\right]$ can be expressed as:

$$Y\left[k\right]=H\left[k\right]·X_{\mathrm{clip}}\left[k\right]+W_{\mathrm{rx}}\left[k\right]$$

(10)

The AWGN $W_{\mathrm{rx}}\left[k\right]$ noise after the FFT operation can be modelled as the zero-mean complex AWGN with two-sided power spectral density of $N_0/2$ per dimension and a variance of ${\sigma }_{w_{\mathrm{rx}}}^2=B{N}_0$ [8]. The embedded pilot sequence is extracted from the received data symbols, then its phase and amplitude are estimated based on the maximum likelihood (ML) algorithm. The estimated pilot phase is used to reverse the phase rotation carried out at the transmitter. The received data symbols are then demodulated before being converted into bit streams followed by calculating the BER of the system based on the decoded binary streams.

3. PAPR Reduction Procedure of PA DCO-OFDM

DCO-OFDM is an optical OFDM variant that requires the addition of the DC bias to convert the negative peaks into a positive signal for intensity modulation. While DCO-OFDM is a spectral efficient modulation technique, it comprises the sum of independent subcarriers in the time-domain which results in the coherent addition of individual subcarriers to produce high PAPR values [20]. Consequently, the high PAPR electrical values require signal clipping at lower and/or upper levels to contain the signal swing within the linear dynamic region of the optical light source [21]. To benefit from the linear dynamic region of the light source in full, the high electrical PAPR peaks must be decreased which helps in minimising the clipping distortion of the transmitted signal [5].

To overcome the PAPR challenge, the PA technique is implemented to reduce the high electrical PAPR peaks of the time-domain signal of the system. In PA, the phase of individual subcarriers is rotated randomly to avoid their coherent addition as much as possible. The implementation process of PA in optical OFDM modulation technique is described in the following section.

The electrical PAPR values of DCO-OFDM in the time-domain is evaluated as: [5]

$$\mathrm{PAPR}=\frac{\underset{0\le n\le N-1}{\max }{\left(\right|x\left[n\right]|}^2)}{\mathrm{E}\left[\right|\mathit{x}\left[\mathit{n}\right]{|}^{\mathit{2}}]}$$

(11)

where $\mathrm{E}[·]$ denotes the statistical expectation. The complementary cumulative distribution function diagram (CCDF) is the most common metric that is used to evaluate the PAPR reduction [5]. The CCDF is defined as the probability that the PAPR of the optical OFDM frame exceeds a predefined reference value.

The steps of reducing the electrical PAPR values of DCO-OFDM using PA algorithm is given as follows [5,22]:

• Group the QAM symbols into U blocks comprising active subcarriers, $N_{\mathrm{subs}}$, to obtain $X^u\left[m\right]$, where $u=1,2,\dots ,U.$ and $m=0,1,\dots ,N_{\mathrm{subs}}-1$.
• Generate multiple iterations of pilot sequence candidates, R, with $N_{\mathrm{subs}}$ length, $X_{\mathrm{p}}^r\left[m\right]$, where $r=1,2,\dots ,R$.
• Select the amplitude, $A_{\mathrm{p}}\left[m\right]$, of the pilot sequence, $X_{\mathrm{p}}^r\left[m\right]$, to be $\pm 1$ only.
• Set the phase value of the pilot sequence ${\theta }_{\mathrm{p}}\left[m\right]$ randomly to 0 or $\pi$ only.
• Rotate the phase of the current DCO-OFDM block, U, by ${\theta }_{\mathrm{p}}\left[m\right]$ of every pilot sequence iteration r.
• Evaluate the PAPR_r of each iteration of $X_{\mathrm{p}}^r\left[m\right]$ in the time domain using (11).
• Select the pilot iteration with the minimum PAPR value, $X_{\mathrm{p}}\left[m\right]=X_{\mathrm{p}}^{\tilde{r}}\left[m\right]$, to be used for PAPR reduction and to be transmitted with the corresponding frames blocks U; where

  $$\tilde{r}=\underset{1\le r\le R}{\mathrm{argmin}{(\mathrm{PAPR}}_r})$$

  (12)
• Embedding the pilot sequence $X_{\mathrm{p}}^{\tilde{r}}\left[m\right]$ into the corresponding frames block, U, and this will make the number of frames per block $\widehat{U}=(U+1)$.

In this manuscript, the high electrical PAPR values of the transmitted time-domain signal are reduced by PA, where the number of frames of DCO-OFDM per block, U, is selected to be $U=5$. This value of U is selected because when the number of U increases, the PAPR of the frames block increases [5]. The number of pilot sequence iterations, R, is selected to be $R=10$ iterations as this value shows a good trade-off between the PAPR reduction gain and computational complexity [15]. At the receiver side, the PA sequences are extracted from all the data blocks and then estimated using ML algorithm which is an optimum detection technique for the statistical parameter, $\theta$, estimation [14]. The ML algorithm is used to estimate the pilot sequence in this work as follows [5]:

The noise-corrupted phase $\widehat{{\theta }_{\mathrm{p}}}\left[m\right]$ of the extracted pilot sequences is estimated using the ML algorithm to improve the recovery of the received data. The estimate of the pilot’s angle is taken between two values only ($\widehat{{\theta }_i}=0$ and $\pi$), that has the minimum Euclidean distance from the phase of the received pilot’s $\widetilde{{\theta }_{\mathrm{p}}}\left[m\right]$. The estimate argument of the angle is defined by the following:

$$\widehat{{\theta }_{\mathrm{p}}}\left[m\right]=\underset{1\le i\le 2}{\mathrm{argmin}}\left[{(\widetilde{{\theta }_{\mathrm{p}}}\left[m\right]-{\theta }_{\mathrm{i}})}^2\right]$$

(13)

The estimated pilot sequence can be given by ${\widehat{X}}_{\mathrm{p}}\left[m\right]=e^{j\widehat{{\theta }_{\mathrm{p}}}\left[m\right]}$, which is equivalent to the following condition [5]:

$${\widehat{X}}_{\mathrm{p}}\left[m\right]=\left\{\begin{array}{cc}+1,\hfill & \mathrm{if}cos\left(\widetilde{{\theta }_{\mathrm{p}}}\left[m\right]\right)\ge 0\hfill \\ -1,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(14)

Additionally, the amplitude of the pilot sequence, $\widehat{A_{\mathrm{p}}}\left[m\right]$, is maintained to unity by using the condition given in (14). The estimated phase, $\widehat{{\theta }_{\mathrm{p}}}\left[m\right]$, of the pilot sequence with its recovered amplitude, $\widehat{A_{\mathrm{p}}}\left[m\right]$, is then applied to estimate and recover the phase of the corresponding subcarrier, m, of the relevant block, U, of the received data symbol.

Figure 3 shows the PAPR reduction gain of PA with 10 iterations. This result quantifies the PAPR reduction gain compared to the basic DCO-OFDM without PAPR reduction. For example, the PA DCO-OFDM system yields almost a 3 dB PAPR reduction gain compared to the conventional DCO-OFDM at a CCDF of ${10}^{-4}$.

4. PA DCO-OFDM Clipping Noise Analysis and Assessment

The clipping of PA DCO-OFDM time-domain signal changes the signal mean and, as a result, its average optical power. Since $x_{\mathrm{clip}}\left[n\right]$ is approximately Gaussian, the nonlinear distortion effects caused by the clipping operation can be modelled according to the Bussgang theorem, and expressed as follows [23]

$$x_{\mathrm{clip}}\left[n\right]=\kappa ·x_{\mathrm{scaled}}\left[n\right]+w_{\mathrm{clip}}\left[n\right]$$

(15)

where $\kappa$ denotes the clipping attenuation factor and $w_{\mathrm{clip}}$ is the nonlinear distortion, i.e., clipping noise imposed by the pre-clipping process and system front-end devices constraints. The nonlinear distortion $w_{\mathrm{clip}}$ is uncorrelated with the transmitted signal $x_{\mathrm{scaled}}\left[n\right]$, $\mathrm{E}[x_{\mathrm{scaled}}\left[n\right]·w_{\mathrm{clip}}\left[n\right]]=0$. The attenuation factor $\kappa$ is a constant that is given as follows [1]:

$$\kappa =\frac{\mathrm{E}[x_{\mathrm{scaled}}\left[n\right]·x_{\mathrm{clip}}\left[n\right]]}{{\sigma }_{x_{\mathrm{scaled}}}^2}$$

(16)

where ${\sigma }_{x_{\mathrm{scaled}}}^2$ is the variance of $x_{\mathrm{scaled}}\left[n\right]$. The nonlinear distortion $w_{\mathrm{clip}}$ is a non-Gaussian noise. However, due to the CLT, its representation in the frequency domain follows a Gaussian distribution with a zero mean and a variance ${\sigma }_{w_{\mathrm{clip}}}^2$. The variance of the clipping noise is given as follows [1,8]:

$$\begin{array}{cc}\hfill \mathrm{E}\left[w_{\mathrm{clip}}\right]& =\mathrm{E}\left[x_{\mathrm{clip}}\right],\hfill \end{array}$$

(17a)

$$\begin{array}{cc}\hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} {\sigma }_{w_{\mathrm{clip}}}^2& =\mathrm{E}\left[w_{\mathrm{clip}}^2\right]-\mathrm{E}{\left[w_{\mathrm{clip}}\right]}^2,\hfill \end{array}$$

(17b)

$$\begin{array}{cc}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \hfill {\sigma }_{w_{\mathrm{clip}}}^2& =\mathrm{E}\left[x_{\mathrm{clip}}^2\right]-{\kappa }^2·{\sigma }_{x_{\mathrm{scaled}}}^2-\mathrm{E}{\left[x_{\mathrm{clip}}\right]}^2.\hfill \end{array}$$

(17c)

In the frequency domain, the clipping noise $w_{\mathrm{clip}}\left[n\right]$ is transformed into additive Gaussian noise according to CLT. Therefore, additive Gaussian noise component with zero-mean and variance ${\sigma }_{w_{\mathrm{clip}}}^2$ is present at each modulated subcarrier in addition to the additive white Gaussian noise $w_{\mathrm{rx}}\left[n\right]$. Therefore, the received signal in the frequency domain can be calculated as follows

$$\begin{array}{cc}\hfill Y\left[k\right]& =H\left[k\right]\left(\kappa ·X\left[k\right]+W_{\mathrm{clip}}\left[k\right]\right)+W_{\mathrm{rx}}\left[k\right],\hfill \\ \\ \hfill & =\kappa H\left[k\right]X\left[k\right]+H\left[k\right]W_{\mathrm{clip}}\left[k\right]+W_{\mathrm{rx}}\left[k\right].\hfill \end{array}$$

(18)

where $Y\left[k\right]$ is the desired received signal, $H\left[k\right]$ is the channel response, $X\left[k\right]$ is the transmitted symbols, $W_{\mathrm{clip}}\left[k\right]$ is the clipping noise, and $W_{\mathrm{rx}}\left[k\right]$ is the additive white Gaussian noise. The received signal can be expressed as the summation of the desired signal, clipping noise, and the receiver noise with zero mean for both noise signals, as shown in (18). The SNR at the kth subcarrier is represented by $\Gamma \left[k\right]$, and can be expressed as follows [8]

$$\Gamma \left[k\right]=\frac{{\kappa }^2\frac{1}{G_{\mathrm{B}}}{\left|H\left[k\right]\right|}^2}{{\sigma }_{w_{\mathrm{clip}}}^2{\left|H\left[k\right]\right|}^2+{\sigma }_{w_{\mathrm{rx}}}^2}$$

(19)

where $G_{\mathrm{B}}$ is the double-sided bandwidth utilisation factor of the optical OFDM frame and ${\sigma }_{w_{\mathrm{rx}}}^2$ is the receiver noise power. The attenuation factor can be simplified from (16) as follows [8]

$$\kappa =Q\left(k_{{\mathrm{b}}_{\mathrm{norm}}}\right)-Q\left(k_{{\mathrm{t}}_{\mathrm{norm}}}\right)$$

(20)

where Q is the Q-function [8]. $k_{{\mathrm{b}}_{\mathrm{norm}}}$ and $k_{{\mathrm{t}}_{\mathrm{norm}}}$ are the normalised bottom and top clipping levels, respectively, and can be expressed as follows [8]

$$k_{{\mathrm{b}}_{\mathrm{norm}}}={\displaystyle \frac{k_{\mathrm{b}}}{{\sigma }_{x_{\mathrm{scaled}}}}}$$

(21)

$$k_{{\mathrm{t}}_{\mathrm{norm}}}={\displaystyle \frac{k_{\mathrm{t}}}{{\sigma }_{x_{\mathrm{scaled}}}}}$$

(22)

The clipping noise variance is given as [8]

$$\begin{array}{cc}\hfill {\sigma }_{w_{\mathrm{clip}}}^2=& \kappa +k_{{\mathrm{b}}_{\mathrm{norm}}}\varphi \left(k_{{\mathrm{b}}_{\mathrm{norm}}}\right)-k_{{\mathrm{t}}_{\mathrm{norm}}}\varphi \left(k_{{\mathrm{t}}_{\mathrm{norm}}}\right)+k_{{\mathrm{b}}_{\mathrm{norm}}}^2\hfill \\ \hfill & [1-Q\left(k_{{\mathrm{b}}_{\mathrm{norm}}}\right)]+k_{{\mathrm{t}}_{\mathrm{norm}}}^{2}Q\left(k_{{\mathrm{t}}_{\mathrm{norm}}}\right)-{\kappa }^2-\hfill \\ \hfill & [\varphi \left(k_{{\mathrm{b}}_{\mathrm{norm}}}\right)-\varphi \left(k_{{\mathrm{t}}_{\mathrm{norm}}}\right)+k_{{\mathrm{b}}_{\mathrm{norm}}}[1-Q\left(k_{{\mathrm{b}}_{\mathrm{norm}}}\right)]+\hfill \\ \hfill & k_{{\mathrm{t}}_{\mathrm{norm}}}Q\left(k_{{\mathrm{t}}_{\mathrm{norm}}}\right){]}^2\hfill \end{array}$$

(23)

where $\varphi (·)$ is the probability density function (PDF) of the standard Gaussian distribution and given by

$$\varphi \left(u\right)=\frac{1}{\sqrt{\left(2\pi \right)}}exp\left(\frac{-u^2}{2}\right)$$

(24)

The complementary cumulative distribution function (CCDF) of the Gaussian distribution $Q(·)$ is given as

$$Q\left(u\right)=\frac{1}{\sqrt{\left(2\pi \right)}}{\int }_u^{\infty }exp\left(-\frac{x^2}{2}\right)dx$$

(25)

The average electrical power of the clipped time-domain signal is given by [8]

$$\begin{array}{cc}\hfill \mathrm{E}\left[{\mathrm{x}}_{\mathrm{clip}}\left[\mathrm{n}\right]\right]=& \varphi \left(k_{{\mathrm{b}}_{\mathrm{norm}}}\right)-\varphi \left(k_{{\mathrm{t}}_{\mathrm{norm}}}\right)+k_{{\mathrm{t}}_{\mathrm{norm}}}Q\left(k_{{\mathrm{t}}_{\mathrm{norm}}}\right)+\hfill \\ \hfill & k_{{\mathrm{b}}_{\mathrm{norm}}}[1-Q\left(k_{{\mathrm{b}}_{\mathrm{norm}}}\right)].\hfill \end{array}$$

(26)

The BER of the kth subcarrier can be approximated as follows [8]

$$\begin{array}{c}\hfill \mathrm{BER}\left[\mathrm{k}\right]\approx \frac{4(\sqrt{M}-1)}{\sqrt{M}{log}_2\left(M\right)}Q\left(\sqrt{\frac{3\times \Gamma \left[k\right]}{M-1}}\right)+\frac{4(\sqrt{M}-2)}{\sqrt{M}{log}_2\left(M\right)}Q\left(3\sqrt{\frac{3\times \Gamma \left[k\right]}{M-1}}\right)\end{array}$$

(27)

5. Results and Discussion

Although the efficacy of the PA technique is proven in [5], the aim here is to show that the presented analysis is valid for predicting the DCO-OFDM-based VLC performance in the presence of clipping noise and PAPR reduction with the PA technique. The performance of PA DCO-OFDM and conventional DCO-OFDM systems are compared in the presence of double-sided signal clipping under AWGN channel in terms of BER and as a function of electrical SNR. The analytical study is verified by simulation for both systems. The two systems are simulated using 4-QAM modulation format and IFFT/FFT size of 2048 with total number of active subcarriers, $N_{\mathrm{subs}}$ = 1023. The optical path gain coefficient, the system bandwidth and the electrical-to-optical conversion coefficient, $\beta$, are constrained to unity in order to prevent any possible noise addition to the system. The optical source is assumed to be an LED with a linear dynamic range between $i_{\min }=5$ mA and $i_{\max }$ = 50 mA, which is assumed to be the source of clipping noise [8]. No clipping at the receiver is assumed. The two systems are compared at an ideal case with no clipping introduced in addition to three clipping cases for analytical and simulation BER performance. In the three clipping cases, the clipping levels are calculated according to the amount of the added bias current, $i_{\mathrm{bias}}$, to the transmitted signal and the LED’s linear dynamic range where, $i_{\mathrm{bias}}$ = 9.8 mA is selected as in [8]. Hence, the bipolar signal is pre-clipped prior to biasing process at bottom level, $k_{\mathrm{b}}=i_{\min }-i_{\mathrm{bias}}$, and at the top level, $k_{\mathrm{t}}=i_{\max }-i_{\mathrm{bias}}$. In the first case (case 1), both signals are clipped at the bottom level, $k_{\mathrm{b}}$ = −4.8 and at the top level, $k_{\mathrm{t}}$ = 40.2. In the second case (case 2), the signals are clipped at $k_{\mathrm{b}}$ = −3.8 and $k_{\mathrm{t}}$ = 41.2, whereas it is clipped at $k_{\mathrm{b}}=-3$ and $k_{\mathrm{t}}$ = 42 in the third case (case 3). These clipping levels are obtained by reducing the $i_{\mathrm{bias}}$ which increases the lower clipping level. The data signal is generated, scaled, clipped, and biased in MATLAB as detailed in Section 2 to form the DCO-OFDM signal. In addition, the same data symbols with the same process and parameters are generated with the PAPR reduction algorithm using $R=10$ iterations to generate the PA DCO-OFDM signal. This process is implemented to investigate the effect of reducing the high PAPR peaks of the signal on the system BER performance. Furthermore, the two signals are compared in terms of BER at the same scaling factor and biasing point for every clipping level. In addition, as a benchmark for comparison with existing results, the ideal case and case 1 for DCO-OFDM in [8] is selected, and signal scaling, biasing, and clipping levels are included. Hence, the average transmitted optical power $P_{\mathrm{opt}}$ = 10 mW. In the ideal case where no clipping is introduced to the transmitted signals, the two systems have achieved identical BER performance, as illustrated in Figure 4a. In clipping case 1, the $i_{\mathrm{bias}}$ is found to be 9.8 mA, and ${\sigma }_{x_{\mathrm{scaled}}}$ = 4.9 mA for DCO-OFDM as obtained in [8]. Whereas, for PA DCO-OFDM, ${\sigma }_{x_{\mathrm{scaled}}}$ = 4.69 mA at the same scaling factor $\alpha$ and $i_{\mathrm{bias}}$. This scaling and biasing setup yield the following normalised clipping levels. In DCO-OFDM, the normalised bottom clipping level $k_{{\mathrm{b}}_{\mathrm{norm}}}$ = −0.98, and the normalised top clipping level $k_{{\mathrm{t}}_{\mathrm{norm}}}$ = 8.2. In PA DCO-OFDM, $k_{{\mathrm{b}}_{\mathrm{norm}}}$ = −1.02, and $k_{{\mathrm{t}}_{\mathrm{norm}}}$ = 8.57. The analytical simulations of this case for the two systems are illustrated in Figure 4b.

In the second case (case 2), the bias current is reduced to $i_{\mathrm{bias}}$ = 8.8 mA, which increases the lower clipping level, and hence, increases the clipping noise. Furthermore, this modifies the normalised clipping levels to be $k_{{\mathrm{b}}_{\mathrm{norm}}}$ = −0.78, and $k_{{\mathrm{t}}_{\mathrm{norm}}}=8.4$ for DCO-OFDM, while for PA DCO-OFDM $k_{{\mathrm{b}}_{\mathrm{norm}}}$ = −0.81, and $k_{{\mathrm{t}}_{\mathrm{norm}}}$ = 8.78. The BER performance of this setup is shown in Figure 4c for both systems.

In the third case (case 3), the normalised clipping levels are modified by reducing the bias current further to $i_{\mathrm{bias}}$ = 8 mA. This results in the increase in the lower-level clipping where $k_{{\mathrm{b}}_{\mathrm{norm}}}=-0.61$, and $k_{{\mathrm{t}}_{\mathrm{norm}}}=8.57$ for DCO-OFDM, while for PA DCO-OFDM, $k_{{\mathrm{b}}_{\mathrm{norm}}}=-0.64$, and $k_{{\mathrm{t}}_{\mathrm{norm}}}$ = 8.95, which are shown in Figure 4d.

In all the clipping cases, the PA DCO-OFDM achieved a better BER performance than that of the conventional DCO-OFDM without the PAPR reduction, as shown in Figure 4b–d. PA DCO-OFDM requires less $i_{\mathrm{bias}}$ to convert the signal’s negative peaks to positive, and hence, PA DCO-OFDM is more efficient in terms of optical power than the conventional DCO-OFDM. Furthermore, PA DCO-OFDM is expected to perform better under clipping conditions, as shown in Figure 4, by reducing the high peaks of the OFDM waveform which results in minimising the clipping distortion and the nonlinearity effect. As a result, system’s SNR is maximised, and hence, a better BER performance of the system is observed. In addition, the PA DCO-OFDM allows the signal to have more swing within the linear dynamic range of the system transmitter and reduces the upper and/or lower level clipping effect. The PA performance improvement reaches its lowest when the clipping noise becomes dominant. The error floor observed in the results is caused by the clipping. And, in all cases, the error floor is lower with PAPR reduction technique implemented. The clipping noise becomes dominant and degrades the performance of both systems. Therefore, the limitation in this work is the clipping noise which causes the BER performance to approach an error floor for both systems.

6. Conclusions

The clipping noise in the PAPR reduced system (PA DCO-OFDM) is investigated analytically in this work. This includes the clipping noise attenuation factor and the variance of the received waveform which are determined in closed-form. The SNR of the system is calculated based on the determined attenuation factor and noise variance. The system’s analytical BER performance of the clipped PAPR reduced PA DCO-OFDM is verified through simulation studies at different clipping levels. The BER of the PA DCO-OFDM is then compared to that of conventional DCO-OFDM without PAPR reduction at the selected clipping levels. The PA DCO-OFDM achieved a better BER performance than that of the conventional DCO-OFDM without PAPR reduction in the all used clipping levels. The high PAPR values reduction in the transmitted signal in PA DCO-OFDM reduces the clipping noise hence, enhances the system BER performance. PAPR reduction has the potential to reduce the transmitted average optical power by reducing the required bias power, $P_{\mathrm{bias}}$, which results in increasing the reliability of the LED and hence, its lifetime span. The BER gain can be used to increase the transmitted optical power for longer transmission distance and/or higher achievable data rate.