1 Introduction

With the promotion of Internet of things, cloud service, virtual reality and other high data-traffic services, radio-based wireless networks are facing increasingly serious congestion problems (Buzzi et al. 2016; Agyapong et al. 2014). As a potential complementary access technology for the 5G wireless systems, optical wireless communication (OWC) gains extensive attention for decades owing to its broad license-free spectrum, high energy efficiency, immunity to electromagnetic interference and so on (Koonen 2017; Chi et al. 2017; Elgala et al. 2011). Generally, intensity modulation and direct detection (IM/DD) system is adopted in cost-sensitive OWC systems, where the instantaneous power of optical emitter is used to carry the information (Wilson and Armstrong 2009). Recently, in IM/DD OWC systems, orthogonal frequency-division multiplexing (OFDM) has been widely investigated owing to its high spectral efficiency, flexible modulation formats, and effective resistance to inter-symbol interference (ISI) (Wang et al. 2015a, b; Tsonev et al. 2014).

Despite its numerous superior characteristics, OFDM systems still face an enormous implementation challenge. The time-domain waveforms of OFDM present an intrinsic defect of high peak-to-average power ratio (PAPR) (Zhou et al. 2015; Zhang et al. 2014). Waveforms with high-PAPR feature are susceptible to nonlinear clipping distortion and quantization distortion, which seriously degrades the system performance (Zhang et al. 2017). Many research efforts have been made for PAPR reduction in OFDM, including amplitude clipping, tone injection, tone reservation, precoding, selected mapping and some hybrid PAPR reduction techniques, which are analyzed and summarized in Han and Lee (2005). Among them, single-carrier FDM (SC-FDM) is well recognized as an outstanding method of PAPR mitigation, which has been deployed commercially in the Long Term Evolution (LTE) network (Ekstrom et al. 2006; Zhang et al. 2010). It employs a discrete Fourier transform (DFT) precoding structure to achieve PAPR reduction without introducing signal distortion and transmission rate loss.

Generally, localized SC-FDM (L-SC-FDM) and interleaved SC-FDM (I-SC-FDM) are two kinds of popular SC-FDM schemes with different subcarrier arrangement strategies (Myung et al. 2006). Compared with the localized one, the interleaved SC-FDM scheme has lower PAPR. Thereby, I-SC-FDM can further relax the requirement to linearity-limited devices such as power amplifier, electro-optic or optic-electro devices, which is economical for the cost-sensitive systems. However, different from L-SC-FDM with the localized subcarrier allocation method, I-SC-FDM can not use Hermitian symmetry to generate the real-valued signal due to the interleaved subcarrier allocation method (Lin et al. 2016). It still restraints I-SC-FDM to implement in IM/DD systems straightforwardly. Fortunately, real-valued I-SC-FDM (RV-I-SC-FDM) has been first proposed where the encoding module at transmitter is greatly simplified in our previous work (Zhou et al. 2017a). Meanwhile, in Zhou et al. (2017b), the RV-I-SC-FDM scheme is applied for achieving the better optical power budgets for short-range optical interconnects. However, to our best knowledge, RV-I-SC-FDM has never been applied or investigated in OWC systems whose free-space channel is different from the fiber channel in optical interconnects.

In this paper, we propose the first RV-I-SC-FDM scheme for mitigating the nonlinearity effect and high-frequency distortion in cost-sensitive OWC systems. The main contributions of this paper are as follows:

  • The novel RV-I-SC-FDM scheme for OWC systems: The computational complexity of the encoding module at transmitter is greatly reduced, which is suitable for cost-sensitive OWC systems. Compared with OFDM, RV-I-SC-FDM has a lower PAPR, making it more immune to the nonlinearity clipping in OWC systems. Meanwhile, for the bandwidth-limited OWC systems, RV-I-SC-FDM has better performance on resistance to the serious high-frequency distortion.

  • The specific and comprehensive analysis of RV-I-SC-FDM system over optical-wireless LOS link and optical-wireless diffuse link: The \(Q^2\)-factor performance and signal-to-noise ratio (SNR) performance on the different subcarriers of RV-I-SC-FDM scheme are analyzed in detail. Furthermore, simulation results show that, under the transmitter nonlinearity and optical-wireless diffuse channel, RV-I-SC-FDM has a significant improvement of the maximum \(Q^2\) compared with conventional OFDM.

2 Principle of I-SC-FDM for OWC system

Fig. 1
figure 1

The system architecture of I-SC-FDM transceiver for OWC systems; a transmitter and b receiver; Subc. Mapper: subcarrier mapper and Subc. Demapper: subcarrier demapper

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Figure 1 shows a system architecture of I-SC-FDM transceiver for OWC systems. At the transmitter side, the randomly-generated binary data is firstly mapped to M-QAM symbols and the M-QAM symbols are then sent into I-SC-FDM encoding module in the blue dashed box. N-point fast Fourier transform (FFT) is employed to achieve the spread operation, which output can be presented as

$$\begin{aligned} X_n= \sum \limits _{k=0}^{N-1}x_k {\mathrm {exp}}\left( \frac{-j2\pi kn}{N}\right) , ~\quad 0 \le n \le N-1 \end{aligned}$$
(1)

where \(x_k\) is the M-QAM symbol. The interleaved mapping manner is used for subcarrier mapper, and the generated data sequence \({\mathbf {X}}{'} ({\mathrm{i.e.}}~[0, X_0, 0, X_1,\ldots , 0, X_{N-1}]\)) is applied to 2N-point inverse FFT (IFFT) operation. Then, the time-domain complex-valued I-SC-FDM (CV-I-SC-FDM) symbol can be achieved:

$$\begin{aligned} x{'}_m= \sum \limits _{l=0}^{2N-1} X{'}_l {\mathrm {exp}} \left(\frac{j2\pi lm}{2N}\right), \quad 0 \le m \le 2N-1. \end{aligned}$$
(2)

By substituting the sequence \({\mathbf {X}}{'}\) and (1) into (2), the expression of \({\mathbf {X}}{'}\) can be simplified as

$$\begin{aligned} x{'}_m= {\left\{ \begin{array}{ll} x_m {\mathrm {exp}}( \frac{j\pi m}{N}) /2, & \quad ~ 0 \le m \le N-1,\\ x_{m-N} {\mathrm {exp}}(\frac{j\pi m}{N})/2, & \quad N < m \le 2N-1. \end{array}\right. } \end{aligned}$$
(3)

Clearly, the signal \({\mathbf {X}}{'}\) is anti-symmetrical, that is \(x{'}_m = -x{'}_{m+N}~(0 \le m \le N-1)\). The useful information is not lost if abandoning the redundant second half with the length of N in CV-I-SC-FDM. Therefore, the encoding part in blue dashed box of Fig. 1 can be substituted with the module in Fig. 2, where \(E_m\) (i.e., \(E_m = {\mathrm {exp}} (j\pi m/N)\)) is the m-th phase rotation. The complexity of the encoding part evaluated by the number of complex multiplications can be greatly reduced from \(O(N{\mathrm {log}}_2 N)\) to O(N) (Zhou et al. 2017a). The significant reduction on computational complexity has great practical significance for cost-sensitive OWC systems.

Fig. 2
figure 2

The schematic diagram of simplified encoding module

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Then, cyclic prefix (CP) is attached to the first-half of CV-I-SC-FDM and complex to real transform (C2RT) is implemented to obtain the RV-I-SC-FDM signal since real-valued signal is required in IM/DD OWC system.

Subsequently, the RV-I-SC-FDM signal is processed by parallel to serial, digital to analog conversion (DAC) and low-pass filter (LPF). In this paper, ideal DAC and LPF are assumed. Finally, after combined with direct current (DC) via bias tee, the generated signal is applied to light-emitting diode (LED). Over optical-wireless channel, the signals are received and processed in the receiver. The decoding part is the opposite process with the encoding one in the transmitter, where channel estimation with least-square (LS) algorithm is implemented to compensate the optical-wireless channel distortion.

3 Channel model for optical wireless communications

In this section, the channel model for OWC is analyzed in detail. Two typical characteristics are present in OWC systems including the limited linear dynamic range of transmitter, e.g. LEDs, and the diffuse transmission of optical signals in free space. In OWC systems, the dominant noise includes the ambient-induced shot noise and electronics-induced thermal noise at receiver, both of which can be modeled as zero-mean and signal-independent additive white Gaussian noise (AWGN) (Carruthers and Kahn 1997; Dimitrov et al. 2012). Then, the received signal in the time-domain can be calculated by

$$\begin{aligned} y(t) = {\mathfrak {R}}\cdot \mathbf {[}x{'}(t)\mathbf {]_{\mathrm {C}}}\otimes h_c(t) + n_{\mathrm {AWGN}}(t) \end{aligned}$$
(4)

where \({\mathfrak {R}}\) denotes the responsivity of the detector, \(h_c(t)\) denotes the impulse response of OWC channel and \(n_{\mathrm {AWGN}}(t)\) denotes the AWGN noise. \(\otimes \) is the convolution operation and \({\mathbf {[\cdot ]_{\mathrm {C}}}}\) denotes the nonlinear-clipping effect of transmitter.

Generally, the simplified double-clipped model is applied to model the limited linear dynamic range of transmitter in OWC systems (Zhang et al. 2014; Komine et al. 2009). The signals beyond the dynamic range of transmitter will be clipped, which leads to the undesired nonlinear clipping distortion. Thereby, we denote the clipped signal as

$$\begin{aligned} {\mathbf {x_c{'}}} = {\left\{ \begin{array}{ll} A_{\mathrm {U}}, & \quad A_{\mathrm {U}}< {\mathbf {x{'}}}+V_{\mathrm {DC}},\\ {\mathbf {x{'}}}+V_{\mathrm {DC}}, & \quad A_{\mathrm {D}} < {\mathbf {x{'}}}+V_{\mathrm {DC}} \le A_{\mathrm {U}},\\ A_{\mathrm {D}}, & \quad A_{\mathrm {D}} \ge {\mathbf {x{'}}}+V_{\mathrm {DC}}, \end{array}\right. } \end{aligned}$$
(5)

where \(V_{\mathrm {DC}}\) is the DC bias, \(A_{\mathrm {U}}\) and \(A_{\mathrm {D}}\) are the upper-clipping bound and lower-clipping bound of signal amplitude, respectively. Then, the linear dynamic range of LED is \([A_{\mathrm {D}}, A_{\mathrm {U}}]~(A_{\mathrm {D}}<A_{\mathrm {U}})\). In order to ensure the stability of illumination and the maximization of modulation depth, \(V_{\mathrm {DC}}\) is set to \((A_{\mathrm {U}}+A_{\mathrm {D}})/2\). Thus, the clipped signal can be revised as

$$\begin{aligned} {\mathbf {x_c{'}}} = {\left\{ \begin{array}{ll} A_{\mathrm {U}}, & \quad \frac{A_{\mathrm {U}}-A_{\mathrm {D}}}{2}< {\mathbf {x{'}}},\\ {\mathbf {x{'}}}+\frac{A_{\mathrm {U}}-A_{\mathrm {D}}}{2}, & \quad \frac{A_{\mathrm {D}}-A_{\mathrm {U}}}{2} < {\mathbf {x{'}}} \le \frac{A_{\mathrm {U}}-A_{\mathrm {D}}}{2},\\ A_{\mathrm {D}}, & \quad \frac{A_{\mathrm {D}}-A_{\mathrm {U}}}{2} \ge {\mathbf {x{'}}}. \end{array}\right. } \end{aligned}$$
(6)

For an explicit description, clipping ratio is employed to indicate the nonlinear-clipping effect of transmit signal, which can be given by

$$\begin{aligned} \zeta _{\mathrm {C}} = \frac{\sigma _{\mathrm {rms}}}{A_{\mathrm {U}}-A_{\mathrm {D}}}, \end{aligned}$$
(7)

where \(\sigma _{\mathrm {rms}}\) denotes the root-mean-square (RMS) value of transmit signal. A larger \(\zeta _{\mathrm {C}}\) is equivalent to the more serious nonlinear distortion signal suffers.

Another crucial feature of OWC system is the diffuse transmission of optical signal in free space. The optical-wireless channel is commonly divided into the line-of-sight (LOS) link and the diffuse link. Frequently, AWGN channel model is used to simulate the LOS link while the optical-wireless diffuse link is modeled with ceiling-bounce (CB) channel model (Carruthers and Kahn 1997) as

$$\begin{aligned} h_c(t) = G_0 \cdot \frac{6\kappa ^6}{(t+\kappa )^7} \cdot \varepsilon (t), \end{aligned}$$
(8)

where \(G_0\) is the optical DC gain, \(\varepsilon (t)\) is the unit step function and the variable \(\kappa \) depends on RMS delay spread \(D_{\mathrm {rms}}\) as \(\kappa = 12 \sqrt{11/13} D_{{\mathrm {rms}}}\).

Based on Eq. (4), the received signal is achieved by convoluting signal with the channel transfer function. For the optical-wireless diffuse link, the delays from the different paths lead to the ISI in the time domain and the frequency-selective fading of signal in the frequency domain (Mohammed et al. 2017). Additionally, Carruthers et al. validated that \(D_{\mathrm {rms}}\) of channel is an accurate predictor of ISI-induced SNR penalty. Therefore, the performance of resistance to ISI will be investigated from the SNR aspect when considering the optical-wireless diffuse link.

4 Simulation results and discussions

Fig. 3
figure 3

a Real part of CV-I-SC-FDM symbol; b imaginary part of CV-I-SC-FDM symbol; c RV-I-SC-FDM symbol and d RV-OFDM symbol

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In this section, simulation results and relevant discussion are present, including time-domain symbol characteristics, PAPR performance and system transmission performance.

4.1 Time-domain symbol characteristics

Figure 3 reveals time-domain characteristics of I-SC-FDM symbol and OFDM symbol. FFT size is set to 64 and 4-QAM is modulated. The green dashed line locates the 33-th sample position. Figure 3a and b show real part and imaginary part of CV-I-SC-FDM symbol, respectively. Obviously, both of them follow anti-symmetrical property. Consequently, RV-I-SC-FDM symbol can be gotten by concatenating first-half of real part and first-half of imaginary part, as illustrated in Fig. 3c. Figure 3d depicts real-valued OFDM (RV-OFDM) symbol with unit power, which commonly is achieved by hermitian symmetry. The power of RV-I-SC-FDM symbol is also normalized to one for fair comparison. Compared with RV-I-SC-FDM symbol, the RV-OFDM symbol has larger peak power. That is, the RV-I-SC-FDM symbol has a lower PAPR than the RV-OFDM symbol.

In the following, the real-valued signals are analyzed and compared on the performance of PAPR and transmission. Thus, the RV-I-SC-FDM symbol and the RV-OFDM signal are denoted as I-SC-FDM symbol and OFDM symbol for short, respectively.

4.2 PAPR performance

Fig. 4
figure 4

Comparison of PAPR performance for I-SC-FDM and OFDM; a 4-QAM and b 16-QAM

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In cost-sensitive OWC systems, PAPR performance is an important factor of affecting the cost of devices, e.g. amplifier, DAC and LED. According to Wang et al. (2016), the PAPR is calculated by

$$\begin{aligned} {\text {PAPR}}= 10 \times {\mathrm {log}}_{10} \left[ \frac{\max \nolimits _{0\le i \le N-1}(|x{'}_i|^2 )}{{ {\mathrm {E}}}(\left| x{'}_i\right| ^2 )}\right] ~({\mathrm { dB}}) \end{aligned}$$
(9)

where \(x{'}_i\) is OFDM signal, and \({\mathrm {E}}(\cdot )\) represents the expected value. For evaluating the effect of PAPR mitigation methods, the cumulative distribution function (CDF) is widely adopted, which denoting the probability of PAPR lower than a threshold value (i.e. \({\mathrm {CDF}} = {\mathrm {P}}_{\mathrm {r}}({\text {PAPR}} \le {\text {PAPR}}_{\mathrm{T}})\)).

Figure 4 presents CDF curves of PAPR for I-SC-FDM and OFDM when 4-QAM and 16-QAM are modulated, respectively. A total of \(10^5\) OFDM symbols are adopted for evaluating PAPR performance and making a comparison in different number of subcarriers (i.e., FFT size = 256, 512 and 1024). For OFDM, the PAPR performance gets worse with the increase of the subcarrier number. In comparison, the number of subcarriers has a negligible effect on the PAPR of I-SC-FDM due to its single carrier characteristics. When the number of subcarriers is set to 256, the PAPR performance of I-SC-FDM decreases by about 10 dB and 6.9 dB compared with OFDM when 4-QAM and 16-QAM are modulated, respectively. Furthermore, the superiority of I-SC-FDM will be more pronounced when the number of subcarriers is increasing.

4.3 System transmission performance

Table 1 Parameters of OWC system configuration
Full size table

In this part, transmission performance of system is investigated by simulation. The detailed parameters of OWC system configuration are shown as Table. 1. The total number of bits is more than \(10^6\) for ensuring that more than 100 error bits can be found for every bit error rate (BER). \(D_{\mathrm {rms}}\) is set to 1.3 ns to simulate the RMS delay of the optical-wireless diffuse channel in a \(5 \times 5 \times 3~({\rm{m}}^{3})\) room (Dimitrov et al. 2012). Training symbols are used for channel estimation via the simple LS algorithm. \(Q^2\) factor is a commonly used metric of transmission performance, which is calculated by

$$\begin{aligned} Q^2 = 20 \times {\mathrm {log}}_{10} [\sqrt{2} \times {\mathrm {ercinv}}(2 \times {\text {BER}})]~({\mathrm {dB}}) \end{aligned}$$
(10)

where \({\mathrm {ercinv}} (\cdot )\) denotes inverse complementary error function.

4.3.1 System transmission performance over optical-wireless LOS link

Fig. 5
figure 5

\(Q^2\) factor comparison between I-SC-FDM and OFDM over optical-wireless LOS link

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For investigating the performance of resisting the transmitter nonlinearity, AWGN channel model is used to simulate the LOS link. Figure 5 reveals that \(Q^2\) factor comparison between I-SC-FDM and OFDM in different SNR over optical-wireless LOS link. For comparing the \(Q^2\) factor performance near the \(7\%\) forward error correction (FEC) limit for 4-QAM and 16-QAM modulated schemes, the relative AWGN power is set to 0.3844 and 0.04 for 4-QAM and 16-QAM, respectively. With the increase of SNR, \(Q^2\) factor is increasing until the performance is affected by nonlinear clipping distortion for I-SC-FDM and OFDM. In comparison, a wide range of performance for I-SC-FDM is above the \(7\%\) FEC limit while OFDM never reaches the limit. The vertices of the four \(Q^2\) curves are marked with olive stars. The maximum \(Q^2\) of I-SC-FDM is about 3.16 dB (2.5 dB) higher than that of OFDM when 4-QAM (16-QAM) is modulated. The corresponding constellations for A, B, C and D points are inserted in Fig. 5. Compared with OFDM, the more centralized constellations are achieved for I-SC-FDM. This is because I-SC-FDM has the lower PAPR, which makes it less susceptible to the transmitter nonlinearity.

4.3.2 System transmission performance over optical-wireless diffuse link

Figure 6 depicts \(Q^2\) versus SNR for I-SC-FDM and OFDM over optical-wireless diffuse link. The CB channel model is used to simulate the optical-wireless diffuse channel. The relative AWGN power is set to 0.04 and 0.0042 for 4-QAM and 16-QAM for comparing the \(Q^2\) factor performance near the \(7\%\) FEC limit, respectively. Like the performance over optical-wireless LOS channel, with the increase of SNR, \(Q^2\) factor is increasing until the performance is affected by the transmitter nonlinearity. The maximum \(Q^2\) of I-SC-FDM is approximately 3.44 dB and 3.14 dB higher than that of OFDM when 4-QAM and 16-QAM are modulated, respectively. Compared with the performance over the optical-wireless LOS channel, the improved \(Q^2\) gains are achieved over the optical-wireless diffuse channel. There exist two reasons for the improved performance. One is the low PAPR of I-SC-FDM, making it more immune to nonlinear clipping distortion. Another reason is the inherent single-carrier characteristics of I-SC-FDM. It can effectively mitigate the high-frequency fading distortion of signal caused by optical-wireless diffuse link, which will be further validated as follows.

Fig. 6
figure 6

\(Q^2\) factor comparison between I-SC-FDM and OFDM over optical-wireless diffuse link

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Taking the 16-QAM modulation as an example, SNRs of each data subcarrier for I-SC-FDM and OFDM are shown in Fig. 7. Error vector magnitude of the received signals is generally used to estimate the SNR (Shafik et al. 2006). The simulation parameters are corresponding to the conditions of C and D points in Figs. 5 and 6, respectively. The purple dash line represents the required SNR (BER = \(7\%\) FEC limit) according to Shafik et al. (2006), which can be calculated by

$$\begin{aligned} {\text {BER}}_0 \approx \frac{2 \times (\sqrt{M} -1)}{\sqrt{M} {\mathrm {log}}_2(\sqrt{M})} Q\left( \sqrt{\frac{6 \times {\mathrm {log}}_2 (\sqrt{M}) {\text {SNR}}}{{\mathrm {log}}_2 M (M-1)}}\right) , \end{aligned}$$
(11)

where \(Q(x) = \int _{ x }^{ \infty } {\frac{1}{\sqrt{2 \pi }} {\mathrm {exp}}(\frac{-\nu ^2}{2}) d\nu }\). When 16-QAM modulation is set, the required SNR is determined for achieving the BER target of FEC limit. According to Eq. (11), \({\text {SNR}} = 15.2\) dB. Over optical-wireless diffuse fading channel, high-frequency components of OFDM suffer from severe power attenuation, thus resulting in the serious deterioration of system performance. For I-SC-FDM, high-frequency distortion can be averaged to all subcarriers owing to the additional N-point FFT spread operation before the odd-subcarrier mapper. The achieved SNR is greater than the required SNR at \(7\%\) FEC limit. Hence, it is robust to the severe high-frequency attenuation. Therefore, the advantage of I-SC-FDM will be more highlighted. This is the reason why the better \(Q^2\) gains are achieved over the optical-wireless diffuse fading channel compared with that over the LOS channel.

Fig. 7
figure 7

SNR versus subcarrier index for I-SC-FDM and OFDM

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5 Conclusions

In this paper, for the first time, we proposed the I-SC-FDM scheme for cost-sensitive OWC systems. The computational complexity of the encoding module at transmitter is reduced from \(O(N {\mathrm {log}}_2 N)\) to O(N), which is suitable for cost-sensitive OWC systems. Compared with OFDM, I-SC-FDM has a lower PAPR, which makes it more immune to the nonlinear clipping in OWC systems. Meanwhile, for the bandwidth-limited OWC systems, I-SC-FDM has better performance on resistance to the serious high-frequency distortion. The transmission performance of this scheme is studied via simulation in detail. Under the transmitter nonlinearity and optical-wireless diffuse fading channel, the maximum \(Q^2\) of I-SC-FDM is 3.44 dB and 3.14 dB higher than that of OFDM when 4-QAM and 16-QAM are modulated, respectively. The results show the feasibility and advantages of I-SC-FDM for cost-sensitive OWC systems.