## Abstract

For orthogonal frequency division multiplexing (OFDM) based indoor visible light communication (VLC) systems, partial non-ideal transmission conditions such as insufficient guard intervals and a dispersive channel can result in severe inter-symbol crosstalk (ISC). By deriving from the inverse Fourier transform, we present a novel time domain reshuffling (TDR) concept for both DC-biased optical (DCO-) and asymmetrically clipped optical (ACO-) OFDM VLC systems. By using only simple operations in the frequency domain, potential high peaks can be relocated within each OFDM symbol to alleviate ISC. To simplify the system, we also propose an effective unified design of the TDR schemes for both DCO- and ACO-OFDM. Based on Monte-Carlo simulations, we demonstrate the statistical distribution of the signal high peak values and the complementary cumulative distribution function of the peak-to-average power ratio under different cases for comparison. Simulation results indicate improved bit error rate (BER) performance by adopting TDR to counteract ISC deterioration. For example, for binary phase shift keying at a BER of 10^{−3}, the signal to noise ratio gains are ~1.6 dB and ~6.6 dB for DCO- and ACO-OFDM, respectively, with ISC of 1/64. We also show a reliable transmission by adopting TDR for rectangle 8-quadrature amplitude modulation with ISC of < 1/64.

© 2017 Optical Society of America

## 1. Introduction

Recently, visible light communication (VLC) using white light-emitting diodes (LEDs) has become an intriguing alternative to the radio frequency (RF) wireless technologies. The key advantages of VLC include unregulated huge bandwidth, high energy efficiency, no electromagnetic interference, multiple functionalities, etc [1,2]. Since current commercial optical components do not allow the phase as a means to convey data information, the intensity modulation with direct detection (IM/DD) scheme is therefore the most widely adopted in VLC due to its simplicity and robustness [3–5]. For this reason, well-studied advanced RF communication techniques are not directly applicable to VLC. There are a number of pulse based modulation schemes that were proposed for VLC [6]. The simplest modulation technique is on–off keying (OOK). With the blue-filtering and equalization techniques, the limited bandwidth of white LED devices can be effectively extended [7,8]. To support dimming control, variable pulse-position modulation (VPPM) was proposed and also mentioned in IEEE 802.15.7 [9]. In addition to the pulse based modulation schemes, color based modulation schemes can be adopted for high data rate applications by utilize the wavelength domain [10]. A promising method taking care of illumination was also proposed in [11]. However, due to hardware imperfections and multi-path effects, VLC still suffers inter-symbol crosstalk (ISC). Orthogonal frequency division multiplexing (OFDM), with its successful implementation in both optical fiber and RF based systems, has been introduced in indoor VLC systems. OFDM can increase the bandwidth efficiency of white LEDs and provide robustness against the multi-path induced dispersion as a result of indoor light reflections [12–17]. There are generally two popular types of IM/DD OFDM schemes: DC-biased optical (DCO-) OFDM and asymmetrically clipped optical (ACO-) OFDM [3–5].

In high-speed indoor VLC environments, OFDM frames from different symbols are likely to interfere with each other whenever the link experiences random multi-path delay, obstruction, user’s mobility, loss of synchronization, etc [14]. For example, in [17] it was observed that higher moving speed tends to result in increased bit error rate (BER) and packet loss rate. OFDM systems are inherently designed with a sufficient guard interval (GI), which could be as long as the channel’s impulse response to cater for frequency domain equalization and bit loading. However, the system data throughput can be reduced if the GI length is too long. Although schemes with shorter GI and with no GI have been reported in optical fiber systems [18,19], there are no such investigations on VLC systems. For indoor VLC systems with insufficient GI, the marginal position of an OFDM frame will still interfere with its neighboring symbols under certain conditions. This is best illustrated in Fig. 1, where the leading edge of the next symbol (shadowed region) overlaps with the current symbol (i.e., a current fast Fourier transform (FFT) window) at the receiver (Rx). This overlap consequently leads to the ISC, which needs to be reduced in order to maintain the required link performance.

The concept of parallel transmission, which is equivalent to extending the symbol length, is the key feature in OFDM systems. However, the drawback is the high peak-to-average power ratio (PAPR) in IM/DD based VLC systems, which can become a major issue [20–23]. In this paper, instead of investigating PAPR reduction techniques, we focus on the distribution of the signal peak values in the time domain for OFDM symbols. Based on conventional DCO- and ACO-OFDM systems, we first demonstrate that for certain types of modulation formats, transmitting signals based on a number of parallel subcarrier channels (SCs) can have higher probability of being combined with a resultant high peak at the leading edge of each OFDM symbol. This means that, the leading edge of an OFDM symbol is more likely to contain a considerable proportion of the signal energy and therefore should be taken care of in order to ensure that it does not severely affect the demodulation performance. To address this problem, we make derivations from the basic formula of inverse Fourier transform (IFT) in detail and propose a time domain reshuffling (TDR) scheme for both DCO- and ACO-OFDM systems. Through simple preprocessing operations in the frequency domain, potential high peaks can be relocated within each symbol without the need of buffering and exchanging long data block sequences in the time domain, thus effectively alleviating the ISC. By analyzing the mechanism of our TDR, we also propose an effective unified design of the TDR schemes for both DCO- and ACO-OFDM systems, which can further simplify the system implementation. The given numerical results verify the correctness of the analysis presented. With Monte-Carlo simulation we show an improvement in the BER performance for a VLC link under a dispersive channel condition.

The rest of the paper is outlined as follow. The principle of the proposed TDR concept for DCO- and ACO-OFDM will be presented in Sections 2 and 3, respectively. The unified design of the TDR scheme will be given in Section 4. Numerical simulation results will be shown in Section 5. And we finally provide a conclusion in Section 6.

## 2. TDR for DCO-OFDM

#### 2.1 Principle

In conventional OFDM systems, the data information is encoded in the frequency domain and transmitted in a number of parallel orthogonal SCs. This is generally implemented by using an inverse FFT (IFFT). The discrete OFDM signal in the time domain is given as [3]:

*S*(

*k*) is the complex symbol mapped on the

*k*th subcarrier channel, and

*N*is the total channel number. For conventional DCO-OFDM, the input of IFFT should satisfy the Hermitian symmetry property for generating the real values. Following IFFT, a DC bias is added to the bipolar real values to generate the strictly positive signal. For a total of 2

_{c}*N*SCs, the Hermitian symmetry property is defined as [4]:

*X*is the complex symbol to be transmitted. Thus, the discrete DCO-OFDM signal in the time domain is expressed by:

_{k}*a*and

_{k}*b*are the real and imaginary parts of

_{k}*X*, respectively. For every time domain sampling point

_{k}*s*(

*n*), while the average value is zero (equivalent to the DC level if considering the DC-bias of the signal), the variance can be given by:

*Var*(

*a*) >

_{k}*Var*(

*b*). For example, for a rectangle 8-QAM with the constellation points in the first quadrant of (1, 1) and (3, 1),

_{k}*Var*(

*a*) and

_{k}*Var*(

*b*) are given as (3

_{k}^{2}+ 1

^{2})/2 and 1

^{2}, respectively. Thus, according to Eq. (4), the variance of

*s*(0) will be larger than the average power of the OFDM signal. Since the amplitude of

*s*(0) is proportional to the cumulative sum of

*a*, therefore it has a higher chance to have a resultant high peak than other positions. Note that the peak signal discussed in this paper is the absolute value, which includes both positive and negative peak values. In the dispersive channel condition, the leading edge of the OFDM symbol will reside outside the FFT window at the Rx, see Fig. 1. Consequently, the link will experience severe ISC if

_{k}*s*(0) happens to be a high peak.

To overcome this problem, we start the derivation from the basic formula of IFT. The IFT of the continuous spectral density function *S*(*ω*) should be written as [29]:

*f*(

*m*) as the discrete-time non-periodic expression of

*f*(

*t*), the inverse discrete-time Fourier transform (IDTFT) corresponding to Eq. (5) can be given as [29]:If

*S*(

*e*) is sampled by 2

^{jω}*N*sampling points with the interval of π/

*N*and denoted by

*S*(

*l*), then from Eq. (6) we have:

*N*−1. However, according to Eq. (7), the range of the sampling points is distributed from −

*N*to

*N*−1. Correspondingly, we rewrite Eq. (2) and then get the following mapping expression as:

*k*=

*l*+

*N*and the time variable

*n*=

*m*+

*N*into Eq. (9) and then we have:

Compared with Eq. (1), Eq. (10) has the extra terms of (−1)* ^{n}* and (−1)

*, which result from different ranges of sampling points between conventional IFFT and Eq. (7). These terms will lead to extra procedures before and after both DCO-OFDM coder and decoder, respectively. In the following part, we will show that the term of (−1)*

^{k}*actually has no effect on the sequence order of*

^{n}*f*(

*n*) in the time domain, so we will just ignore it. However, we will focus on the significant term of (−1)

*because it reshuffles OFDM symbols in the time domain, which can alleviate the ISC under the non-ideal transmission conditions. Therefore, based on Eq. (10), we propose a new TDR scheme for the DCO-OFDM system as illustrated in Fig. 2.*

^{k}The operation principle is the same as the conventional scheme, except for the preprocessing procedure prior to IFFT at the transmitter (Tx), which is achieved by simply changing the sign of the odd SCs while keeping the even SCs unchanged. For the proposed scheme, the generated discrete OFDM signal can be given as:

At the Rx reverse procedure posterior to FFT is adopted to ensure full recovery of the original data as in the conventional system. Actually, by adopting such simple operations in the frequency domain, each OFDM symbol is equivalently exchanging its first half with the second half in the time domain, thus the *time domain reshuffling* concept, which can be represented as:

*in Eq. (10).*

^{n}#### 2.2 Comparison module in DCO-OFDM Systems with TDR

From Eq. (12), the potential high peaks at the symbol leading edge can be moved to the middle part of the symbol by means of reshuffling (i.e., more specifically the targeted sampling point *s*(0) is exchanged with *s*(*N*)). However, it is not difficult to show that *s*(*N*) is also proportional to the cumulative sum of (−1)* ^{k}a_{k}*. This means that, SCs still tend to add up to a high peak at

*s*(

*N*). When

*s*(

*N*) carries more energy than

*s*(0), the link will still experience severe ISC if only the simple reshuffling scheme is used. In this work, we propose to include a comparison module for the DCO-OFDM system with TDR, see Fig. 2. By comparing

*s*(0) with

*s*(

*N*), reshuffling is conducted selectively to ensure that the actual higher peak resides within each symbol. Specifically, reshuffling is conducted only when we have:

## 3. TDR for ACO-OFDM

In conventional ACO-OFDM systems, the data information is only mapped to the odd IFFT inputs. Compared with DCO-OFDM, all even inputs are set to zero. Considering a total of 4*N* SCs for ACO-OFDM, the arrangement of the complex symbols on SCs should be given as [4]:

Starting from Eq. (6), if *S*(*e ^{jω}*) is sampled by 4

*N*sampling points with the interval of π/(2

*N*) and denoted by

*S*(

*l*), then from Eq. (6) we have:

*N*−1. However, according to Eq. (15), the range of the sampling points is distributed from −2

*N*to 2

*N*−1. Correspondingly, we rewrite Eq. (14) and then get the following mapping expression as:

*f*(

*m*) is also reduced by half due to lack of sampling in Eq. (16). Considering the actual range of IFFT for ACO-OFDM, variable substitution is needed for Eq. (17). Therefore, we simultaneously substitute both the frequency variable

*k*=

*l*+

*N*and the time variable

*n*=

*m*+

*N*into Eq. (17) and then we have:

*N*SCs are considered for ACO-OFDM, the zero SCs should be inserted into Eq. (18). Obviously,

*f*(

*n*) can be further written as:

Similar as DCO-OFDM, we will only focus on the significant term of (−*j*)* ^{k}*. Based on Eq. (19), we propose a new TDR scheme for the ACO-OFDM system as illustrated in Fig. 3.

The procedure is similar to the TDR scheme for DCO-OFDM, where the even SCs are kept unchanged and the odd SCs are multiplied by the (−*j*)* ^{k}* sequence. The generated discrete OFDM signal can actually be given as:

*s*(0) is exchanged with

*s*(3

*N*). In Section 5, we will show that high peaks are less likely to be present in

*s*(3

*N*) compared to

*s*(0) in ACO-OFDM. Although using a comparison module will improve the system performance to a certain degree, we do not include it in ACO-OFDM for consideration of the system work load and spectral efficiency.

The schematic diagrams of the TDR scheme proposed for DCO- and ACO-OFDM symbols in Figs. 2 and 3 are shown in Fig. 4, respectively. Note that we only focus on analyzing the high peaks at the symbol leading edge. This is due to the ISC arising at the symbol leading edge under the partial non-ideal transmission conditions such as insufficient GI and a dispersive channel, as shown in Fig. 1. Therefore, in Fig. 4, if high peaks occur in the middle part of OFDM symbols, there is no need to conduct TDR, because these high peaks will not be distorted due to ISC.

## 4. Unified design of TDR

Conventionally, in order to reshuffle the OFDM signal, we may need complex procedures such as buffering and exchanging the long data block sequence in the time domain. However, by adopting parallel preprocessing for the SCs in the frequency domain, we can simply reshuffle the OFDM signal without the need of buffering and exchanging procedures, which saves both storage and time. However, from the above analysis, the TDR schemes for DCO- and ACO-OFDM systems have various implementation approaches, respectively. This is due to the different results derived from the basic formula of IFT. Through comparison between the two schemes, there are still significant similarities which can further help to optimize the implementation of reshuffling.

For the DCO-OFDM system with a total of *N _{c}* SCs, the preprocessing procedure of TDR can be seen as multiplying SCs by the (−1)

*sequence in the frequency domain. Actually, (−1)*

^{k}*can be denoted as*

^{k}*exp*(

*j*2π∆

*nk*/

*N*) with ∆

_{c}*n*to be

*N*/2. According to the time shifting property of inverse discrete Fourier transform (IDFT) given by [33]:

_{c}*sequence can lead to a time shifting by*

^{k}*N*/2 for the sampling points of the DCO-OFDM symbol. This is in accordance with the TDR for the DCO-OFDM symbol in Fig. 4. On the other hand, for the ACO-OFDM system with a total of

_{c}*N*SCs, reshuffling is realized by means of multiplying SCs by the (−

_{c}*j*)

*sequence in the frequency domain. Actually, (−*

^{k}*j*)

*can be denoted as*

^{k}*exp*(

*j*2π∆

*nk*/

*N*) with ∆

_{c}*n*to be −

*N*/4. According to Eq. (22), this explains the time shifting effect of the ACO-OFDM symbol in Fig. 4.

_{c}In Fig. 5, based on the time shifting property of IDFT, we propose a unified design of the TDR schemes for both DCO- and ACO-OFDM systems, which can be seen as a generalization of the TDR schemes in Section 2 and Section 3.

The operation principle is similar as the conventional scheme, except for the preprocessing procedure prior to IFFT at the Tx, which is achieved by simply multiplying all SCs by the *exp*(*j*2π∆*nk*/*N _{c}*) sequence. The generated discrete OFDM signal can be given as:

*n*can be adjusted to achieve the required time shift. As a result, the reshuffling of each OFDM symbol can be represented as:

Note that most commonly used communication systems are designed to transmit equiprobable random data streams. This also means that, in VLC systems the transmitted data bits are usually evenly distributed. Therefore, the above proposed TDR schemes are mainly based on the assumption of evenly distributed random data streams. However, as for other kinds of random data streams, as long as high peaks tend to appear at the leading edge of OFDM symbols, TDR can always relocated them to within each symbol. Other kinds of random data streams may be adopted in some cases, but they are perhaps out of the scope of this current paper.

## 5. Results and discussions

In this section, we will evaluate the effect of reshuffling on indoor VLC systems under the partial non-ideal transmission conditions. The unified TDR scheme proposed in Fig. 5 is mainly adopted in both DCO- and ACO-OFDM systems. A data source with a pseudo-random binary sequence, which consists of equiprobable “0”s and “1”s, is used to encode and modulate parallel SCs. For each figure, more than 5*10^{4} of OFDM frames are tested in Monte-Carlo simulations. Modulation formats with *Var*(*a _{k}*) >

*Var*(

*b*) are considered. However, for modulation formats such as quadrature phase shift keying (QPSK) or 4-QAM, because

_{k}*Var*(

*a*) =

_{k}*Var*(

*b*), adopting reshuffling will not lead to performance improvement based on the discussion in Section 2. Therefore, provided

_{k}*Var*(

*a*) >

_{k}*Var*(

*b*), reshuffling is an effective solution to overcome the ISC. Note that although the reshuffling concept is proposed for certain modulation formats with

_{k}*Var*(

*a*) >

_{k}*Var*(

*b*), this will not limit its futuristic application. The reasons are given as follows. Firstly, there have been a number of modulation formats with the property of

_{k}*Var*(

*a*) >

_{k}*Var*(

*b*) [24–28] and many other novel schemes are emerging. These modulation formats will potentially be adopted in indoor OFDM VLC systems. Secondly, the bit and power loading schemes of SCs at the Tx have been proposed in OFDM VLC systems [34,35] due to their bright future as in the RF wireless systems. When simultaneously transmitting BPSK, QPSK and other signals using different SCs we will have

_{k}*Var*(

*a*) >

_{k}*Var*(

*b*), therefore reshuffling can still be used in such hybrid VLC systems. Thirdly, for the case of

_{k}*Var*(

*a*) ≤

_{k}*Var*(

*b*), the proposed TDR will not cause performance degradation at all. Since the TDR scheme only requires simple operations in the frequency domain, the complexity is very low. From a system implementation point of view, such TDR design is a promising approach for futuristic system extension when it comes to the case of

_{k}*Var*(

*a*) >

_{k}*Var*(

*b*). In this paper, we only focus on investigating BPSK and rectangle 8-QAM, respectively. BPSK is the simplest modulation format for OFDM VLC SCs. In addition, the constellation points of BPSK only contain the in-phase component

_{k}*a*while the quadrature component

_{k}*b*is zero. Therefore, at first we can focus on investigating the performance of TDR when only considering the effect of the in-phase component variation for BPSK. Then we take into account the effect of the quadrature component. For QPSK or 4-QAM,

_{k}*Var*(

*a*) =

_{k}*Var*(

*b*). Therefore, considering

_{k}*Var*(

*a*) >

_{k}*Var*(

*b*) discussed in our proposed TDR scheme, rectangle 8-QAM is the simplest feasible QAM format that contains both the in-phase and quadrature components. Please note that the quadrature component of rectangle 8-QAM is non-zero, which is different from BPSK. Therefore, we can choose to use rectangle 8-QAM to extend the research scope to QAM modulation formats. Since this paper aims at a proof-of-concept, other higher modulation formats and hybrid modulation schemes adopting bit and power loading will be the subject of future investigations.

_{k}In Figs. 6 and 7, the distributions of the top 5% high peaks for DCO- and ACO-OFDM signals in the time domain with and without reshuffling are illustrated, respectively. We adopt a total of 5*10^{4} OFDM symbols to obtain each distribution. Take Fig. 6(d) as an example, for 8-QAM ACO-OFDM with 32 SCs, the total size of binary data stream is 3*(32/4)* (5*10^{4}). Here we only show the representative 5% case with a total of 32 SCs, and note that if the top 1% and 10% high peaks or more SCs are considered, a similar distribution will be observed. Also note that the conceptual design of TDR is based on a statistical perspective. Different random data streams will be encoded and modulated into different OFDM symbols and the transmitted waveform is randomly unpredictable. However, the probabilities of the high peaks generated at different positions can be predictable. From Figs. 6(a)-6(d), SCs of the conventional OFDM signal exactly synchronize at the discrete positions of *s*(0) and *s*(16). High peaks have higher probabilities to occur here than any other positions. Besides, it can be also easily found that *s*(8) and *s*(24) have lower probabilities to form high peaks. In order to effectively reduce the total ISC, we can select to relocate and exchange *s*(0) with a sampling point whose probability of high peaks is as small as possible. Therefore, according to Fig. 6, the optimum time shift ∆*n* for the TDR scheme adopted in Fig. 5 should be either *N _{c}*/4 or 3

*N*/4, which can simultaneously relocate potential high peaks within each OFDM symbol and ensure the lowest signal energy at the symbol leading edge for all the random data streams. Without loss of generality, we assume ∆

_{c}*n*to be

*N*/4 for both DCO- and ACO-OFDM in the rest of this paper. Therefore, with inclusion of TDR, OFDM symbols tend to have high peaks at the positions of

_{c}*s*(8) and

*s*(24) than any other positions, see from Figs. 7(a) − 7(d). More specifically,

*s*(0) is relocated and exchanged with

*s*(8). This is owing to the

*exp*(

*j*π

*k*/2) sequence multiplied in the preprocessing procedure. As a result, with the low-energy

*s*(8) at the symbol leading edge, the ISC can be effectively alleviated under partial non-ideal transmission conditions. Note that for DCO-OFDM we cannot make sure if TDR really reduces ISC for all the random data streams every time without the comparison module. However, it works for most random data streams. For example, numerical results show that, by adopting TDR in Fig. 5, about 76% actual higher peaks at

*s*(0) can be relocated to within each OFDM symbol for BPSK with a total of 32 SCs. Figures 7(e) and 7(f) also illustrate the cases when adopting the TDR scheme in Fig. 2 with the comparison module for DCO-OFDM. It can be shown that with the potential high peaks much more concentrated at

*s*(16), the symbol leading edge also have lower possibilities to form high peaks so as to alleviate ISC. Owning to the comparison module, we can make sure that

*s*(0) and

*s*(16) will be compared every time and the actual higher peak will be relocated within each DCO-OFDM symbol. Moreover, due to the periodicity of SCs,

*s*(0) and

*s*(16) will not simultaneously show high peaks. Therefore, ISC can be effectively reduced by TDR for all the random data streams in this case. In the following, the scheme in Fig. 2 will also be evaluated as a contrast to verify if the comparison module can bring extra BER performance improvement in the DCO-OFDM system.

Figures 8(a) and 8(b) depict simulation results for the complementary cumulative distribution functions (CCDFs) of PAPR against the PAPR for DCO- and ACO-OFDM, respectively for a range of IFFT. The commonly used definition of CCFD can be found in [36]. From the basic point of view, BPSK is taken as an example in Fig. 8. Similar results can be also observed for rectangle 8-QAM. We adopt a total of 1*10^{5} OFDM symbols to obtain each curve. The plots in red represent CCDFs including all time domain samples, which is discrete especially for a small number of SCs. The discrete high PAPR is due to only a few discrete high peak values of OFDM signals. For larger SC numbers, there are an increasing number of discrete peak values, which contribute to the continuous profiles of the plots. In order to explain the reason for these high peak values, we also show plots in blue to represent CCDFs, which exclude *s*(0) and *s*(*N _{c}*/2) for both DCO-OFDM and ACO-OFDM, respectively. By comparison, CCDF plots without these sampling points are much smoother and closer to the vertical axis. This means that, the partial PAPR is exclusively reduced. Thus confirming that the potential high peaks at

*s*(0) and

*s*(

*N*/2) of OFDM symbols do make a significant contribution to the high PAPR in conventional OFDM VLC systems. Therefore, the TDR method can be used to effectively rearrange the positions of potential high peaks in order to alleviate ISC. Note that the purpose of TDR is to reduce ISC by relocating high peaks at the symbol leading edge instead of eliminating them. Therefore, with TDR high peaks still exist but at the new positions within OFDM symbols. Also note that the proposed TDR scheme can still be effectively adopted when considering the nonlinear LED effect. For the high peaks at the symbol leading edge, if they are clipped or distorted due to nonlinear LED effect, then their amplitudes are still likely to be larger than that at other positions. Therefore, TDR is still required to reduce ISC in the presence of nonlinear LED effect, which will be investigated in the upcoming work.

_{c}When the VLC link experiences random multi-path delay, obstruction, user’s mobility, loss of synchronization, etc., the received signal will be dispersive, and OFDM symbols are likely to interfere with each other. However, due to the case of insufficient GI or non-GI, imperfect channel knowledge at the Rx will invalidate conventional channel estimation schemes, which results in severe ISC. Since this paper mainly aims at a proof-of-concept of the proposed TDR scheme, therefore, based on the fundamental analysis we have assumed the scenarios as shown in Fig. 1. In this way we can simply focus on investigating the impact of losing the potential high peaks at the symbol leading edge under the partial non-ideal transmission conditions. And we can also clearly show the effectiveness of all the TDR schemes for further comparison and evaluation. In the following simulation of BER performance, we assume ISC accounting for 1/64, 1/32, 3/64 and 1/16, respectively, of the OFDM symbol length in a back-to-back VLC system with the additive white Gaussian noise at the Rx. The scenario is shown in Fig. 9. The shadowed regions of ISC include the symbol leading edge containing potential high peaks and should be protected by means of TDR. The performance of TDR considering a specific real indoor VLC multi-path dispersive channel will be experimentally investigated in the future work.

Figure 10 depicts the BER against the signal to noise ratio (SNR) for BPSK and rectangular 8-QAM DCO- and ACO-OFDM with and without TDR, respectively, for a total of 64 SCs. The TDR scheme in Fig. 5 is adopted with the optimum ∆*n* to be *N _{c}*/4. We adopt a total of 1*10

^{5}OFDM symbols to obtain each BER point by Monte-Carlo simulation. This ensures that at least 100 error bits are obtained for each BER point in order to decrease the random fluctuation. In Figs. 10(a) and 10(b), the insufficient GI is assumed to account for 1/64 of the OFDM symbol length, whereas in Figs. 10(c) and 10(d), there are no GI. For DCO-OFDM with BPSK and with ISC of more than 1/32, we can observe the asymptotic BER performance due to ISC, thus making it very challenging to achieve the forward error correction (FEC) BER limit of 10

^{−3}[37]. Utilizing the TDR method can result in an improvement in BER performance. For example, at a BER of 10

^{−3}, a ~1.6 dB of SNR gain can be obtained for ISC of 1/64. For ACO-OFDM with BPSK, the asymptotic BER performance can be also observed for all cases with no TDR. By adopting TDR for ACO-OFDM, the system BER can be effectively improved. At a BER of 10

^{−3}, the SNR gain is ~6.6 dB for ISC of 1/64. The improvement is more significant for ACO-OFDM because of doubling of ISC during the demodulation process due to the symmetry property of ACO-OFDM. When using a rectangle 8-QAM, both DCO- and ACO-OFDM systems without TDR cannot support reliable communications. By adopting TDR, we can still observe asymptotic BER plots for all cases. However, with ISC of less than 1/64, a BER of 10

^{−3}can be eventually achieved for ACO-OFDM systems.

In order to verify whether the comparison module in Fig. 2 can bring extra BER performance improvement for DCO-OFDM, Fig. 11 depicts the BER against the SNR for BPSK and rectangular 8-QAM DCO-OFDM with the TDR scheme in Fig. 2 and Fig. 5, respectively, for a total of 64 SCs. Other parameters are kept consistent with Fig. 9. For DCO-OFDM with BPSK, adopting the comparison module can achieve a slightly reduced BER with ISC of less than 1/32. When ISC accounts for more than 3/64, deteriorated BER performance will be observed with the comparison module. Therefore, the TDR scheme in Fig. 5 should be preferred in general for DCO-OFDM with BPSK. However, for DCO-OFDM with rectangle 8-QAM, Fig. 10(c) have shown that although the TDR scheme in Fig. 5 can alleviate ISC to a certain degree, it still does not work for a reliable transmission because of lack of comparison module. Furthermore, adopting the TDR scheme in Fig. 2 can achieve better BER performance for all cases, as shown in Fig. 10(b). Specifically, when ISC is less than 1/64, reliable communications with a BER of 10^{−3} can be eventually achieved for DCO-OFDM systems using rectangular 8-QAM with adoption of the comparison module. Therefore, for DCO-OFDM with rectangle 8-QAM, the TDR scheme in Fig. 2 should be preferred. Figure 11(b) actually shows the necessity of adopting the comparison module. Note that, if ISC accounts for much less than 1/64, e.g., 1/128 or 1/256, improved BER performance is expected, which will be shown in future publications.

Finally, Fig. 12 depicts the constellation diagrams with and without TDR, respectively, for a total of 64 SCs and a SNR at 15 dB. The normalization of signal power is also assumed. Since the modulation of ACO-OFDM is similar as that of DCO-OFDM except for the zero value of even inputs, we only focus on the constellation diagrams of DCO-OFDM in this paper. For BPSK and rectangle 8-QAM, the ISC is assumed to account for 1/16 and 1/64 of the OFDM symbol length, respectively. Results show that the quality of recovered signal can be improved significantly by adopting TDR. This also proves the effectiveness of the comparison module in Fig. 2. It is worth to mention that for rectangle 8-QAM OFDM, since only the first sampling point *s*(0) is assumed to be distorted by ISC, therefore the resulted noise mainly appears in the signal real part due to the property of FFT, as shown in Fig. 12(c).

## 6. Conclusion

In indoor VLC systems based on OFDM, we first showed that the SCs tend to compose resultant high peaks at the leading edge of each OFDM symbol when using certain types of modulation formats with variance of the real part being larger than the imaginary part. This makes the systems more vulnerable to the ISC under partial non-ideal transmission conditions. In order to relocate the potential high peaks at the OFDM symbol leading edge, we made derivations from the basic formula of IFT in detail and accordingly proposed a TDR scheme for both DCO- and ACO-OFDM systems. A unified design of TDR was further proposed for the system simplification. The proposed TDR scheme is attractive because it can reshuffle the OFDM signal in the time domain by only using low-complexity operations in the frequency domain. Monte-Carlo simulation results showed an improvement in the BER performance for the VLC link under a dispersive channel condition. For BPSK, at a BER of 10^{−3} and with the ISC of 1/64 the SNR gains were ~1.6 dB and ~6.6 dB for DCO-OFDM and ACO-OFDM, respectively. We also demonstrated a reliable transmission by adopting TDR using rectangle 8-QAM with the ISC of less than 1/64.

## Funding

National Natural Science Foundation of China (NSFC) (61271239).

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