Underwater visible light communication at 3.24 Gb/s using novel two-dimensional bit allocation

Peng Zou; Yiheng Zhao; Fangchen Hu; Nan Chi

doi:10.1364/OE.390718

1. Introduction

Visible light communication (VLC) is a type of wireless communication that has many advantages, such as its sizeable potential spectrum, cost-effectiveness, and proper confidentiality. Furthermore, it has no electromagnetic interference and is license-free [1–3]. Owing to the weak attenuation of visible light from 450 nm to 550 nm in seawater, underwater VLC (UVLC) based on blue and green spectra has promising applications in submarine tasks such as oil exploration, water quality monitoring, submarine communication, and others [4,5].

Thanks to the low cost and high electrical conductivity of Si material, the Si-substrate based GaN LED has been widely utilized in the UVLC system to achieve high data rate transmission [6]. However, the achievable data rate of the UVLC system is always limited by the bandwidth of commercial light-emitting diodes (LEDs), which is only 15–25 MHz [7,8]. To overcome this difficulty, a high-speed UVLC system can be obtained by expanding the available modulation bandwidth and increasing the spectral efficiency. Hardware equalization based on a T-bridge passive resonant circuit is one of the methods to expand the modulation bandwidth of an LED from several tens of megahertz to 400–500 MHz [9–11]. For a UVLC channel with severe nonlinear effect and ISI, multi-carrier modulation such as orthogonal frequency division multiplexing (OFDM) and discrete multi-tone (DMT) can enhance the system source entropy (SE) through bit-loading algorithms according to their subcarrier signal-to-noise ratio (SNR) and preset forward error correction (FEC) threshold [12–16]. Bit loading algorithms are processed by utilizing the channel SNR at the transmitter and the receiver. The bit and power allocation of the multi-carrier modulation system can generally be divided into two cases: rate maximization and margin maximization [17–20]. The rate maximization algorithms aim at obtaining the highest spectrum efficiency with the target power budget [15,21], while the goal of margin maximization algorithms is to transmit the target bits with the least power budget [22,23]. The Levin-Campello (LC) algorithm, which performs rate maximization or margin maximization, can achieve high speed transmission with the preset power budget for the integer bit-loading multi-carrier modulation system [24,25]. However, the concealed condition behind the LC algorithm is that the SNR of each subcarrier can be adjusted by the allocated power linearly. If the system operates at nonlinear regime, the SNR cannot be adjusted linearly, and the power allocation algorithms cannot meet the target. Unfortunately, when the UVLC system operates at high signal peak to peak voltage (V_pp) and large bandwidth, which are two necessary conditions for high-speed transmission, the UVLC system suffers from high nonlinear distortion [26]. This characteristic of UVLC systems sharply deteriorates the performance of power allocation algorithms.

Hybrid modulation has emerged as a tradeoff between complexity and performance. The most popular hybrid modulation schemes involve time-domain hybrid modulation [27], which uses the SNR gap between traditional QAM and the Shannon limit by concatenating two strings of QAM symbols with adjacent QAM orders [28–30]. The complexity of time-domain hybrid modulation is much lower than the state-of-the-art shaping technology probabilistic shaping [31], which is beneficial for the realization of a practical system. However, few studies have been conducted on frequency and time domain combined hybrid modulation in multi-carrier modulation systems such as DMT. Chen et al. have proposed the precoding-assisted TDHQ with uniform power allocation, which can achieve performance comparable with the LC algorithm and a lower peak to average power ratio [32]. However, the performance of TDHQ is strongly affected by the subcarrier SNR distribution. If some subcarriers are in a deep fading state, the system performance deteriorates dramatically because the precoding technology will decrease the overall SNR according to the principle of the proposed precoding technology.

In this study, to improve the spectral efficiency with low computational complexity when nonlinearity applies, we propose a novel two-dimensional bit allocation (2DBA) algorithm in the DMT UVLC system. The bit error rate (BER) and data rate performance of different modulation schemes including conventional LC, 2DBA, floor bit loading (FBL), and TDHQ are investigated in a blue LED based UVLC system over 1.2 m underwater transmission. Experimental results and analysis verified that 2DBA is more suitable for the high nonlinear effect and ISI distorted UVLC system. The contributions of this paper are summarized as follows:

(1) This study marks the first time that a 2DBA hybrid modulation scheme is proposed in a UVLC system.
(2) When the signal bandwidth is large, or V_pp is high, which means the system suffers from high ISI or high nonlinear effect, this paper proved by analysis and experiment that 2DBA outperforms the LC and TDHQ algorithms.
(3) The data rate of 3.24 Gb/s is achieved by pre-emphasis assisted 2DBA respectively. As far as we know, this is the highest data rate reported in the blue LED chip–based UVLC system.

2. Principle

2.1. SNR and BER estimation

First, we measure the system SNR of each subcarrier. The SNR estimation of the i^th subcarrier in DMT bit-loading can be expressed as

(1)$${SN}{{R}_i} = \frac{{\frac{1}{{{N_t}}}\sum\limits_{j = 0}^{{N_t} - 1} {{{||{S_{i,j}^{{tx}}} ||}^2}} }}{{\frac{1}{{{N_t}}}\sum\limits_{j = 0}^{{N_t} - 1} {{{||{S_{i,j}^{{tx}} - S_{i,j}^{{rx}}} ||}^2}} }}, $$

where N_t is the total number of QAM symbols on each subcarrier, $S_{i,j}^{{tx}}$ is the j^th transmission (TX) QAM symbol on the i^t^h subcarrier, and $S_{i,j}^{\textrm{rx}}$ is the j^t^h receiver (RX) QAM symbol on the i^t^h subcarrier. According to the SNR estimated by Eq. (1), the BER of QAM symbols can be estimated by the SNR. For 2²ⁿ (n = 0,1,…) QAM symbols, the BER estimated by SNR of each subcarrier can be expressed as [33]

(2)$$\Phi (M,\textrm{SN}{\textrm{R}_i}) \approx \frac{{2(1 - \frac{1}{{\left\lceil {\sqrt M } \right\rceil }})}}{{{{\log }_2}(\left\lceil {\sqrt M } \right\rceil )}} \cdot (\frac{1}{2}\textrm{erfc}(\sqrt {\frac{{3{{\log }_2}(\left\lceil {\sqrt M } \right\rceil )}}{{({{\left\lceil {\sqrt M } \right\rceil }^2} - 1) \cdot {{\log }_2}(M)}}\textrm{SN}{\textrm{R}_i}} )), $$

where M is the QAM order and $M = {2^{2k}}$. Additionally, $\textrm{erfc}(x)$ is the complementary Gaussian error function, and [34]

(3)$$\textrm{erfc}(x) = \frac{2}{{\sqrt \pi }}\int_x^\infty {{e^{ - {t^2}}}} dt. $$

For the $M = {2^{2k + 1}}$ QAM symbols except for $k = 1$, the BER of cross QAM constellations can be accurately predicted by [35]

(4)$$\Phi (M,\textrm{SN}{\textrm{R}_i}) = (\frac{{{G_P} \cdot {N_n}}}{{{{\log }_2}(M)}}) \cdot 0.5 \cdot \textrm{erfc}(\sqrt {\frac{{48 \cdot \textrm{SN}{\textrm{R}_i}}}{{31 \cdot M - 32}}} ), $$

where ${G_P}$ is the Gray penalty and ${N_n}$ is the average number of nearest neighbors for one symbol in the constellation and$M = {2^{2k + 1}}$. As 8QAM is specially designed and many geometrical shaped topologies are proposed, we choose the “diamond” topology as our 8QAM constellation. For the 8QAM constellation, the BER can be estimated by [35]:

(5)$$\Phi (M,\textrm{SN}{\textrm{R}_i}) = \frac{{10}}{{16}} \cdot \textrm{erfc}(\sqrt {\frac{{48 \cdot \textrm{SN}{\textrm{R}_i}}}{{31 \cdot M - 32}}} ). $$

According to Eqs. (2)–(5), we can obtain the mapping relationship between the minimum required SNR and the QAM order below the preset FEC threshold. Table 1 summarizes the results at different FEC thresholds.

Table 1. Minimum SNR (dB) for a specific QAM order at different FEC thresholds

View Table

According to the results shown in Table 1, we can proceed with the bit allocation by the estimated SNR of each subcarrier and allocate QAM symbols on each subcarrier. In this paper, we define the FEC threshold as ${\Phi _{th}} = 0.0038$, which is the 7% overhead HD-FEC.

2.2. Principles of 2DBA, LC, FBL, and TDHQ

The bits per QAM symbol (BPS) of the MCM system can be expressed as

(6)$${b_i} = {\log _2}\left( {1 + \frac{{{p_i} \cdot {h_i}}}{{{\Gamma _i} \cdot \sigma_i^2}}} \right), $$

where ${p_i}$ is the average power of the i^th subcarrier, ${h_i}$ is the channel frequency response, ${\sigma _i}$ is the noise variance, and ${\Gamma _i}$ is the SNR loss between the estimated SNR and the practical SNR with a performance below the preset FEC threshold. Since it is difficult to estimate ${h_i}$ and $\sigma _i^2$ directly, we usually use Eq. (1) to estimate the channel SNR. The SNR is measured by transmitting quadratic phase-shift keying (QPSK) symbols on each subcarrier whose ${p_i} = 1$. The FEC threshold is not considered and ${\Gamma _i} = 1$. Therefore, Eq. (6) can be rewritten as

(7)$${b_i} = {\log _2}\left( {1 + \frac{{{p_i} \cdot \textrm{SN}{\textrm{R}_i}}}{{{\Gamma _i}}}} \right). $$

From Eqs. (6) and (7), we can conclude that if the noise variance ${\sigma _i}$ and channel frequency response ${h_i}$ do not change with the power level ${p_i}$, we can utilize power allocation to adjust the SNR distribution between different subcarriers linearly. However, if the noise variance ${\sigma _i}$ and channel frequency response ${h_i}$ change with the power level ${p_i}$, the performance of power allocation will be significantly distorted.

The bit allocation in the frequency domain, that of the i^th subcarrier in the time domain, and the power allocation in the frequency domain of FBL [6,36] are shown in Figs. 1(b), 1(f), and 1(j), respectively. The red lines in Figs. 1(a)–1(i) are the adaptive SE of the estimated SNR that satisfies the FEC threshold. The principle of FBL is illustrated as

(8)$$\begin{array}{{c}} \begin{array}{c} {b_{j,i}} = \max {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\log _2}({M_i}),{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} i = 0,1,\ldots ,{N_f} - 1,{\kern 1pt} {\kern 1pt} j = 0,1,\ldots ,{N_t} - 1\\ {H_i} = {b_{j,i}},j = 0,1,\ldots ,{N_t} - 1 \end{array}\\ {s.t.{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {p_i} = 1,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} i = 0,1,\ldots ,{N_f} - 1}\\ \begin{array}{l} \Phi ({M_i},{p_i} \cdot \textrm{SN}{\textrm{R}_i}) \le {\Phi _{th}} = \Phi \left( {{M_i},\frac{{{p_i} \cdot \textrm{SN}{\textrm{R}_i}}}{{{\Gamma _i}}}} \right)\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\log _2}({M_i}) \ge 1 \end{array} \end{array}, $$

where ${b_{j,i}}$ is the SE on the i^th subcarrier of the j^th DMT symbol, ${N_f}$ is the total number of subcarriers in the frequency domain, ${N_t}$ is the total number of DMT symbols in a frame, ${p_i}$ is the power allocation on the i^th subcarrier, ${M_i}$ is the QAM order on the i^th subcarrier, and ${H_i}$ is the SE on the i^th subcarrier. Although the FBL algorithm can achieve a stable BER performance below the specific FEC threshold, there is a big gap between the system channel capacity and the practical SE. Therefore, the channel capacity cannot be fully utilized.

Fig. 1. Modulation schemes for 2DBA, FBL, LC, and TDHQ: (a)–(d) SE of 2DBA, FBL, LC, and TDHQ in the frequency domain; (e)–(h) SE of the i^th subcarrier of 2DBA, FBL, LC, and TDHQ in the time domain; (i)–(l) power level of 2DBA, FBL, LC, and TDHQ in the frequency domain.

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For LC, the bit allocation in the frequency domain, that of the i^th subcarrier in the time domain, and the power allocation in the frequency domain are shown in Figs. 1(c), 1(g), and 1(k), respectively. The algorithm details are shown in [24]. The goal of the VLC bit allocation system is to achieve the highest SE performance with a preset power budget (generally determined by V_PP) below the preset FEC threshold. Therefore, the LC algorithm in the UVLC system should satisfy two conditions, which are named as efficient and E-tight [25]. The flow of conducting LC algorithm in this paper is listed as follow:

(1). Initialize the power of each subcarrier as 1. The total bit number goal as B=0. The total number of valid subcarriers is${N_f}$. Therefore, the total power budget E_T is ${N_f}$.

(2). Measure the SNR of each subcarrier by QPSK symbols.

(3). Round off to the nearest QAM order for each subcarrier according to Table. 1.

(4). Set the total bit number goal B to the sum of bit numbers on each subcarrier in step 3.

(5). Execute the LC algorithm to reach the total bit number goal B condition and efficient condition.

(6). Calculate the power on each subcarrier.

(7). If the E-tight goal is not reached, B = B+1. Return to (5). Else end up, return B, the power allocation, and bit allocation of each subcarrier.

Figures 1(c), 1(g), and 1(k) show the adjusted bit allocation and power after processing with the LC algorithm. However, the LC algorithm needs considerable computational complexity to meet the E-tight condition [37]. Besides, the order of QAM symbols on the i^th subcarrier in the time domain is fixed. The channel capacity of each subcarrier is just enough to support the data transmission at the specific FEC threshold. Although the LC algorithm can make full use of the channel capacity with integer bits, the algorithm relies on the condition that the power is proportional to the channel SNR. If the system operates in the nonlinear region, or the noise power density changes with the allocated power, increasing or decreasing the power cannot adjust the SNR to the preset value.

The bit allocation in the frequency domain, that of the i^th subcarrier in the time domain, and the power allocation in the frequency domain of TDHQ are shown in Figs. 1(d), 1(h), and 1(l), respectively. The principle of this algorithm is shown in [32]. The SNR of all the valid bandwidth is equalized by precoding (such as discrete Fourier transform spread (DFTS)), which can be expressed by

(9)$$\textrm{SN}{\textrm{R}_e} = \frac{{\sum\limits_{i = 0}^{{N_f} - 1} {{p_i}} }}{{\sum\limits_{i = 0}^{{N_f} - 1} {{p_i}/\textrm{SN}{\textrm{R}_i}} }}. $$

Here ${p_i}\textrm{ = }1$ for any i from 0 to ${N_f} - 1$. The precoding method can reduce the system peak to average power ratio (PAPR) to some degree and improve the system performance. However, if some subcarriers are in deep fading, the system performance deteriorates severely. For example, considering a channel that has 11 subcarriers, in which the SNR on each subcarrier is [25,196,329,424,481,500,481,424,329,196,25], respectively (which behaves like a quadratic convex function), then $SN{R_e} = 102.65$, which is much lower than the average SNR of 310. From Eq. (7), we can know that the average SNR decides the channel capacity. Therefore, the DFTS-assisted TDHQ cannot achieve as much channel capacity as bit allocation or power loading. Besides, the tremendous noise power of the deep fading channel deteriorates the overall system performance. As a consequence, TDHQ may not be an optimal solution to the UVLC system; this is proved in Section. 4.

For 2DBA, the bit allocation in the frequency domain, the bit allocation of the i^th subcarrier in the time domain, and the power allocation in the frequency domain is shown in Figs. 1(a), 1(e), and 1(i), respectively. The proposed 2DBA algorithm is a two-dimensional hybrid modulation scheme, which distributes two adjacent order QAM symbols on the same subcarrier in the time domain. For the i^th subcarrier, we write the higher order as ${Q_{2,i}}$ and lower order as ${Q_{1,i}}$, the number of ${Q_{2,i}}$ QAM symbols in a frame as ${N_{2,i}}$, and the number of ${Q_{1,i}}$ QAM symbols in a frame as ${N_{1,i}}$. The principle of 2DBA is illustrated as follows:

(10)$$\begin{array}{{c}} {{b_{j,i}} = {Q_{2,i}}{\kern 1pt} {\kern 1pt} or{\kern 1pt} {\kern 1pt} {\kern 1pt} {Q_{1,i}},{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} i = 0,1,\ldots ,{N_f} - 1}\\ {{b_{j,i}} = \left\{ {\begin{array}{{l}} {{Q_{1,i}},{\kern 1pt} {\kern 1pt} j = 0,1,\ldots {N_{1,i}}}\\ {{Q_{2,i}},{\kern 1pt} {\kern 1pt} {\kern 1pt} j = {N_{1,i}} + 1,..{N_t} - 1} \end{array}} \right.}\\ {{H_i} = \frac{{{{\log }_2}({Q_{1,i}}) \cdot {N_{1,i}} + {{\log }_2}({Q_{2,i}}) \cdot {N_{2,i}}}}{{{N_t}}}}\\ {{p_i} = 1{\kern 1pt} ,{\kern 1pt} {\kern 1pt} i = 0,1,\ldots ,{N_f} - 1}\\ {s.t.{\kern 1pt} {\kern 1pt} {\Phi _{th}} = \frac{{{{\log }_2}({Q_{1,i}}) \cdot {N_{1,i}} \cdot \Phi ({Q_{1,i}},{p_i} \cdot \textrm{SN}{\textrm{R}_i}) + {{\log }_2}({Q_{2,i}}) \cdot {N_{2,i}} \cdot \Phi ({Q_{2,i}},{p_i} \cdot \textrm{SN}{\textrm{R}_i})}}{{{N_{1,i}} \cdot \log 2({Q_{1,i}}) + {N_{2,i}} \cdot \log 2({Q_{2,i}})}}}\\ {{N_{1,i}} + {N_{2,i}} = {N_t}}\\ {{N_{1,i}} > 0,{\kern 1pt} {\kern 1pt} {\kern 1pt} {N_{2,i}} > 0,{\kern 1pt} {\kern 1pt} {Q_{1,i}}{\kern 1pt} \ge 1{\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {Q_{2,i}} \ge 1} \end{array}. $$

As Figs. 1(a), 1(e), and 1(i) show, the 2DBA fills the SNR gap between FBL and the channel capacity by hybrid modulation below ${\Phi _{th}}$ for each subcarrier. Furthermore, as we utilize the same power allocation scheme as that of SNR estimation, which is the same as the procedure of estimating SNR, the system SNR does not change very much. As fluctuation exists in the UVLC system, we set a “safety gap” $\gamma $ between the calculated SE and the practical SE, which is the number of ${Q_2}$ QAM symbols in a frame whose order reduces to ${Q_1}$. Therefore, when we obtain the value of ${N_1}$ and ${N_2}$ from Eq. (10), the practical symbol number ${N_1}$ and ${N_2}$ can be expressed as

(11)$$\left\{ {\begin{array}{{c}} {{N_1} = N,{N_2} = 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ,{N_2} \le \gamma N}\\ {{N_1} = {N_1} + \gamma N,{N_2} = {N_2} - \gamma N{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ,{N_2}{\kern 1pt} > \gamma N} \end{array}} \right.. $$

In this paper, $\gamma $ is set to 10%, which results in a decrease of 0.1 bit/symbol SE on each subcarrier. As a trade-off, this operation provides a more stable BER performance for the UVLC system. The revised 2DBA SE and that of FBL at different SNR (dB) are shown in Fig. 2. As Fig. 2(a) shows, we also listed the SE of TDHQ, FBL and ceil bit loading (CBL, The SE of CBL is one bit higher than FBL at each SNR) as a comparison. The TDHQ adjusts the power of higher order QAM to realize an optimal SE at each SNR. Therefore, the SE of TDHQ grows almost linearly with the SNR. However, for the case of 2DBA, the power ratio is fixed to 1, thus the SE grows nonlinear with SNR due to the fact of the nonlinear transfer function between BER and SNR. It can be found that the SE of 2DBA is slightly lower than that of TDHQ at each given SNR. In the following sections, we have verified by theoretical analysis and experimental results that this kind of nonlinear SE loading scheme can provide better nonlinear resilience for the UVLC system. In the following experiment, we adjust the SE of each subcarrier according to the revised values shown in Fig. 2.

Fig. 2. (a). SE of TDHQ, 2DBA, FBL, and CBL. (b). BER and SNR relationship of different QAM

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To equalize the average SE and BER performance of DMT symbols in a frame, we introduce a simple interleaving rule to change the mapping of the ${Q_2}$ QAM symbols and ${Q_1}$ QAM symbols. The interleaving rule is illustrated in Fig. 3. If we map the QAM symbols as Fig. 3(a) shows, the SE of the last DMT symbol in a frame is much larger than that of the first DMT symbol, as the blue line in Fig. 3(c) depicts. This results in a huge difference between the BER of the first and last DMT symbols. The mapping rule of TDHQ is cyclically shifting the frame in the time domain [32], in this way the output PAPR can be reduced. In the case of 2DBA, in order to avoid high PAPR, the mapping rule is flipping over the frame in the time domain for only the even subcarriers, and the odd subcarriers are kept unchanged, as illustrated in Figs. 3(a) and 3(b). After the interleaving procedure, 2DBA can present a relatively static SE in the time domain. TDHQ persists of only two modulation formats, and the power ratio between them should be adjusted for a given SNR. However, for the case of 2DBA, 8 different modulation choices can be coexisted, i.e., from BPSK to 256QAM, and the power of different QAM is kept the same.

Fig. 3. Mapping rules of interleaved QAM symbols for 2DBA. (a). Not interleaved. (b). Interleaved. (c) SE versus DMT symbol number

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The proposed 2DBA combines the high spectrum efficiency of bit allocation and hybrid modulation without the complex power allocation procedures. When the UVLC system operates in a state with wide bandwidth and strong nonlinearity, 2DBA outperforms the LC, TDHQ, and FBL schemes; this is analyzed in Section 2.3.

The steps of 2DBA can be summarized as follows:

(1) Calculate the channel SNR of each subcarrier by QPSK signals.
(2) Calculate ${N_1}$, ${N_2}$, ${Q_1}$, and ${Q_2}$ for each subcarrier according to Eq. (10).
(3) If ${Q_1} = 8$, then set ${N_1}\textrm{ = }N$, ${N_2} = 0$, and ${Q_2} = 8$.
(4) Revise ${N_1}$ and ${N_2}$ by Eq. (11).
(5) Interleave the symbols on each subcarrier.
(6) Modulate the symbols by DMT and save ${N_1}$, ${N_2}$, ${Q_1}$, and ${Q_2}$ for demodulation.

2.3. Nonlinear effect and ISI distortion on power allocation

The GaN based LED has a relatively slow carrier lifetime. Therefore, when we increase the bandwidth of the electrical driving signal, the inter-symbol interference of the optical signal will become severe due to the limited carrier recovery time. As the UVLC system has a strong nonlinear response from the input time domain signal to the output time domain signal, the signal will suffer from nonlinear distortion induced by adjacent symbols. To analyze the impact of the nonlinear effect and ISI of the time domain signal on the channel, establishing a precise model of the UVLC system is necessary. However, the model includes many components, including LED, PIN diode, UVLC channel, and electrical amplifier. It is difficult to build a model comprising all the characteristics of these components. Therefore, we choose to build a simple UVLC system model that includes the nonlinear effect of the LED and ISI for our analysis. The model is depicted in Fig. 4. The amplitude magnitude (AM) response is the normalized amplitude response between the output signal after inverse fast Fourier transform (IFFT) and the input signal before fast Fourier transform (FFT), which can be represented by a quadratic polynomial [38,39]. The ISI exists in the UVLC system as well. The frequency response and noise power on the k^th subcarrier are written as $H(k)$ and ${N_0}(k)$, respectively. Here we assume the noise power on each subcarrier will not change.

Fig. 4. VLC modeling for nonlinear effect and ISI. ATT: attenuator; PA: power allocation; AM: amplitude magnitude response; ISI: inter-symbol interference; AWGN: additive white Gaussian noise; ZF: matched zero-forcing receiver.

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For the DMT system, the QAM symbol $X(k)$ on the k^th subcarrier ($k = 0,1,\ldots ,{N_f} - 1$), the time domain symbol $s(t)$ after IFFT can be written as

(12)$$s(t) = \sum\limits_{k = 0}^{{N_f} - 1} {S(k){e^{j\frac{{2\pi }}{{{N_f}}}kt}}} \textrm{ = }\sum\limits_{k = 0}^{{N_f} - 1} {P(k)H(k)X(k){e^{j\frac{{2\pi }}{{{N_f}}}kt}}}. $$

For a more concise expression, the conjugated part of the DMT signal is ignored in the following analysis, which is necessary to ensure that the time domain signal is a real value. Furthermore, if power allocation (PA) is invalid, such as in FBL and 2DBA, the $P(k)$ for any k is equal to 1. The following situations are analyzed in this paper:

(1) Without additive white Gaussian noise (AWGN), AM, and ISI:

The received signal in the time domain can be written as

(13)$$r(t)\textrm{ = }s(t). $$

Furthermore, the received signal in the frequency domain can be depicted as

(14)$$R(k) = \sum\limits_{t = 0}^{{N_f} - 1} {r(t){e^{ - j\frac{{2\pi }}{{{N_f}}}kt}}} = H(k)X(k). $$

For a matched zero-forcing (ZF) receiver, the decoded symbol of the ${k^{\textrm{th}}}$ subcarrier is

(15)$$Y(k) = {H^{ - 1}}(k)H(k)X(k) = X(k). $$

Because $Y(k) = X(k)$, which means the signal is perfectly demodulated, the SNR on the ${k^{\textrm{th}}}$ subcarrier is

(16)$$\textrm{SN}{\textrm{R}_k} = \frac{{\frac{1}{{{N_t}}}\sum\limits_{k = 0}^{{N_t} - 1} {||{X(k)} ||} }}{{\frac{1}{{{N_t}}}\sum\limits_{k = 0}^{{N_t} - 1} {||{Y(k) - X(k)} ||} }} \to + \infty. $$

(2) With AWGN only, without AM and ISI:

As the noise power density on each subcarrier for a UVLC system is different, in this paper, the noise power density on each subcarrier is allowed to vary. The received signal in the time domain can be written as

(17) $$r(t) = s(t). $$

The received signal in the frequency domain can be depicted as

(18)$$R(k) = \sum\limits_{t = 0}^{{N_f} - 1} {r(t){e^{ - j\frac{{2\pi }}{{{N_f}}}kt}}} = H(k)X(k) + \sqrt {{N_0}(k)} \cdot \varepsilon (k), $$

where $\varepsilon (k)$ is a random number with a mean value of zero and a variance of one. Then the signal after ZF can be written as

(19)$$\begin{array}{l} Y(k) = {H^{ - 1}}(k)H(k)X(k)\textrm{ + }{H^{ - 1}}(k)\sqrt {{N_0}(k)} \cdot \varepsilon (k)\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = X(k)\textrm{ + }{H^{ - 1}}(k)\sqrt {{N_0}(k)} \cdot \varepsilon (k) \end{array}. $$

The SNR of the received signal is written as

(20)$$\begin{array}{l} \textrm{SN}{\textrm{R}_k} = \frac{{\frac{1}{{{N_t}}}\sum\limits_{i = 0}^{{N_t} - 1} {{{||{X(k)} ||}^2}} }}{{\frac{1}{{{N_t}}}\sum\limits_{i = 0}^{{N_t} - 1} {{{||{Y(k) - X(k)} ||}^2}} }}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = \frac{{\frac{1}{{{N_t}}}\sum\limits_{i = 0}^{{N_t} - 1} {{{||{X(k)} ||}^2}} }}{{\frac{1}{{{N_t}}}\sum\limits_{i = 0}^{{N_t} - 1} {{{\left||{{H^{ - 1}}(k)\sqrt {{N_0}(k)} \cdot \varepsilon (k)} \right||}^2}} }} \end{array}. $$

If PA is valid, Eq. (18) is rewritten as

(21)$$R(k) = \sum\limits_{t = 0}^{{N_f} - 1} {r(t){e^{ - j\frac{{2\pi }}{{{N_f}}}kt}}} = \sqrt {P(k)} H(k)X(k) + \sqrt {{N_0}(k)} \cdot \varepsilon (k). $$

Then the signal after ZF is written as

(22)$$Y(k) = X(k) + {(\sqrt {P(k)} )^{ - 1}}{H^{ - 1}}(k)\sqrt {{N_0}(k)} \varepsilon (k). $$

Therefore, the SNR on the k^th subcarrier after PA is

(23)$$\begin{array}{l} \textrm{SN}{\textrm{R}_k} = \frac{{\frac{1}{{{N_t}}}\sum\limits_{i = 0}^{{N_t}\textrm{ - }1} {{{||{X(k)} ||}^2}} }}{{\frac{1}{{{N_t}}}\sum\limits_{i = 0}^{{N_t}\textrm{ - }1} {{{\left||{{{(\sqrt {P(k)} )}^{ - 1}}{H^{ - 1}}(k)\sqrt {{N_0}(k)} \cdot \varepsilon (k)} \right||}^2}} }}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = \frac{{P(k) \cdot \frac{1}{{{N_t}}}\sum\limits_{i = 0}^{{N_t}\textrm{ - }1} {{{||{X(k)} ||}^2}} }}{{\frac{1}{{{N_t}}}\sum\limits_{i = 0}^{{N_t}\textrm{ - }1} {{{\left||{{H^{ - 1}}(k)\sqrt {{N_0}(k)} \cdot \varepsilon (k)} \right||}^2}} }} \end{array}. $$

Comparing with Eq. (20), the SNR with PA is just $P(k)$ times that without PA.

(3) With AWGN and AM, without ISI:

If AM response of the time domain signal is considered, the received signal without PA in the time domain can be written as

(24)$$r(t) = {b_1}s(t) + {b_2}{(s(t))^2} = {r_1}(t) + {r_2}(t), $$

The typical value of ${b_1}$ and ${b_2}$ are 1.5354 and -0.5333 respectively. Therefore, the nonlinear distortion introduced by the second-order term cannot be ignored. The ${r_2}(t)$ is the intermodulation products (IMP) term, which can be expressed as [40]

(25)$${r_2}(t) = \sum\limits_{p = 1}^{{N_f} - 1} {\sum\limits_{q = 1}^{{N_f} - 1} {S(p)S(q){e^{j\frac{{2\pi }}{{{N_f}}}(p + q)t}}} } + \sum\limits_{p = 1}^{{N_f} - 1} {\sum\limits_{q = 1}^{{N_f} - 1} {S(p){S^\ast }(q){e^{j\frac{{2\pi }}{{{N_f}}}(p - q)t}}} }. $$

Then the received signal in the frequency domain can be depicted as

(26)$$\begin{array}{l} R(k) = \sum\limits_{t = 0}^{{N_f} - 1} {r(t){e^{ - j\frac{{2\pi }}{{{N_f}}}kt}}} \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = {b_1}H(k)X(k) + \sum\limits_{t = 0}^{{N_f} - 1} {{r_2}(t){e^{ - j\frac{{2\pi }}{{{N_f}}}kt}}} + \sqrt {{N_0}(k)} \cdot \varepsilon (k)\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = {b_1}H(k)X(k) + {b_2}SX(k) + \sqrt {{N_0}(k)} \cdot \varepsilon (k) \end{array}, $$

where $SX(k)$ is the IMP term of the k^th subcarrier in the frequency domain and

(27)$$SX(k) = \sum\limits_{i = 1,j = k - i}^{k - 1} {S(i)S(j) + 2\sum\limits_{i = 1,j = k - i}^{k - 1} {S(i){S^\ast }(j)} }. $$

Considering other terms as interference terms, the signal after ZF can be expressed as

(28)$$Y(k) = X(k) + \frac{{{b_2}}}{{{b_1}}}{H^{ - 1}}(k)SX(k) + \frac{1}{{{b_1}}}{H^{ - 1}}(k)\sqrt {{N_0}(k)} \cdot \varepsilon (k). $$

Therefore, the SNR of the k^th subcarrier can be expressed as

(29)$$\begin{array}{l} \textrm{SN}{\textrm{R}_k} = \frac{{\frac{1}{{{N_t}}}\sum\limits_{i = 0}^{{N_t} - 1} {{{||{X(k)} ||}^2}} }}{{\frac{1}{{{N_t}}}\sum\limits_{i = 0}^{{N_t} - 1} {{{||{Y(k) - X(k)} ||}^2}} }}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = \frac{{\frac{1}{{{N_t}}}\sum\limits_{i = 0}^{{N_t} - 1} {{{||{X(k)} ||}^2}} }}{{\frac{1}{{{N_t}}}\sum\limits_{i = 0}^{{N_t} - 1} {{{\left||{\frac{{{b_2}}}{{{b_1}}}{H^{ - 1}}(k)SX(k) + \frac{1}{{{b_1}}}{H^{ - 1}}(k)\sqrt {{N_0}(k)} \cdot \varepsilon (k)} \right||}^2}} }} \end{array}. $$

When PA is valid, Eq. (26) can be expressed as

(30)$$\begin{array}{l} R(k) = \sum\limits_{t = 0}^{{N_f} - 1} {r(t){e^{ - j\frac{{2\pi }}{{{N_f}}}kt}}} \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = {b_1}\sqrt {P(k)} H(k)X(k) + \sum\limits_{t = 0}^{{N_f} - 1} {{r_2}(t){e^{ - j\frac{{2\pi }}{{{N_f}}}kt}}} + \sqrt {{N_0}(k)} \cdot \varepsilon (k)\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = {b_1}\sqrt {P(k)} H(k)X(k) + {b_2}SX\_P(k) + \sqrt {{N_0}(k)} \cdot \varepsilon (k) \end{array}, $$

where the IMP term of the k^th subcarrier is

(31)$$SX\_P(k) = \sum\limits_{i = 1,j = k - i}^{k - 1} {\sqrt {P(i)P(j)} S(i)S(j) + 2\sum\limits_{i = 1,j = k - i}^{k - 1} {\sqrt {P(i)P(j)} S(i){S^\ast }(j)} }. $$

Then the signal after ZF can be expressed as

(32)$$Y(k) = X(k) + \frac{{{b_2}}}{{{b_1}}}{P^{ - 1}}(k){H^{ - 1}}(k)SX\_P(k) + \frac{1}{{{b_1}}}{P^{ - 1}}(k){H^{ - 1}}(k)\sqrt {{N_0}(k)} \varepsilon (k). $$

Therefore, the SNR of the k^th subcarrier with PA can be expressed as

(33)$$\begin{array}{l} \textrm{SN}{\textrm{R}_k} = \frac{{\frac{1}{{{N_t}}}\sum\limits_{k = 0}^{{N_t} - 1} {{{||{X(k)} ||}^2}} }}{{\frac{1}{{{N_t}}}\sum\limits_{k = 0}^{{N_t} - 1} {{{||{Y(k) - X(k)} ||}^2}} }}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = \frac{{P(k) \cdot \frac{1}{{{N_t}}}\sum\limits_{k = 0}^{{N_t} - 1} {{{||{X(k)} ||}^2}} }}{{\frac{1}{{{N_t}}}\sum\limits_{k = 0}^{{N_t} - 1} {{{\left||{\frac{{{b_2}}}{{{b_1}}}{H^{ - 1}}(k)SX\_P(k) + \frac{1}{{{b_1}}}{H^{ - 1}}(k)\sqrt {{N_0}(k)} \cdot \varepsilon (k)} \right||}^2}} }}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{ = }P(k) \cdot \textrm{SNR}{\_\textrm{L}_k} \cdot \frac{{\frac{1}{{{N_t}}}\sum\limits_{k = 0}^{{N_t} - 1} {{{\left||{\frac{{{b_2}}}{{{b_1}}}{H^{ - 1}}(k)SX(k) + \frac{1}{{{b_1}}}{H^{ - 1}}(k)\sqrt {{N_0}(k)} \cdot \varepsilon (k)} \right||}^2}} }}{{\frac{1}{{{N_t}}}\sum\limits_{k = 0}^{{N_t} - 1} {{{\left||{\frac{{{b_2}}}{{{b_1}}}{H^{ - 1}}(k)SX\_P(k) + \frac{1}{{{b_1}}}{H^{ - 1}}(k)\sqrt {{N_0}(k)} \cdot \varepsilon (k)} \right||}^2}} }} \end{array}, $$

where $\textrm{SNR}\_{\textrm{L}_k}$ is the SNR in Eq. (30). According to Eqs. (29), (31), and (33), we can easily recognize that the IMP terms distort the SNR with PA and AM response.

(4) With AWGN, AM, and ISI:

The received signal in the time domain can be expressed as

(34)$$r(t) = \sum\limits_{\tau ={-} {t_\tau }}^{{t_\tau }} {{w_\tau }} ({b_1}s(t + {t_\tau }) + {b_2}{(s(t + {t_\tau }))^2}). $$

Here $2{t_\tau } + 1$ is the number of taps of ISI, and ${w_\tau }$ is the weight of the tap $\tau $. It is easy to conclude that the SNR cannot be adjusted by PA to the target value in the system with active ISI and nonlinear distortion.

From the above analysis, we can conclude that as nonlinear AM response and ISI exist in the UVLC system, the performance of PA is distorted severely. At the same time, the SNR measured by QPSK is still valid for 2DBA and FBL. Regarding the noise model in optical communication and wireless communication, a more generalized and simplified model usually assumes the noise power to be constant when varying the frequency. However, for the UVLC system, the noise power density is highly dependent on the received light intensity [41]. Therefore, the performance of conventional PA will further deteriorate.

3. Experimental setup

The experimental setup is depicted in Fig. 5. The process of DMT mapping and DMT demapping (DMT Demap.) blocks is similar to that in [42]. First, we estimate the channel SNR based on QPSK data. The procedure of SNR estimation by QPSK is similar to that of FBL except for the mapping and demapping block. For channel SNR estimation, all the subcarriers are mapped by QPSK symbols. After the procedures of DMT mapping, the time domain DMT signals are loaded by an arbitrary waveform generator (AWG). At the hardware transmitting side (TX) in Fig. 5, the AWG transmits the signals at a preset sample rate and signal V_pp. Then, the signal emitted by the AWG is equalized by a T-bridge hardware passive circuit that can expand the signal bandwidth from 10–20 MHz to 550 MHz. Afterward, the signal is amplified and coupled with the DC to generate a non-negative signal, which is necessary for the intensity modulation and direct detection (IM-DD) UVLC system. The non-negative signal is loaded by a blue silicon substrate LED, which can convert electrical signals into an optical signal. Subsequently, after the signal is transmitted underwater for 1.2 m, it is received by the PIN diode of the receiving side.

Fig. 5. Experimental setup of SNR estimation based on 2DBA, TDHQ, LC, and FBL; S/P: serial to parallel; AWG: arbitrary waveform generator; Eq: hardware equalizer; EA: electrical amplifier; PIN: p-i-n junction diode; RX: received message; TIA: trans-impedance amplifier; OSC: oscilloscope.

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At the hardware receiving side (RX) in Fig. 5, the optical signal detected by a PIN is converted into a current signal, and the weak current signal is converted to the voltage signal by the trans-impedance amplifier (TIA). The hardware low-pass filter removes high-frequency noise from the signal, and the amplified voltage signals are sampled by using an oscilloscope. Then, the signal is loaded by the receiving offline processing platform. Afterward, the DMT demapping process converts the received time domain signal into frequency domain QAM symbols, and the SNR estimation block calculates the SNR of each subcarrier by error vector magnitude (EVM) and saves the subcarrier SNR value.

For 2DBA, the QAM orders and symbol numbers on each subcarrier are derived from the subcarrier SNR and Eq. (11). Then, the binary bits are allocated on each subcarrier and are mapped into the corresponding QAM ${Q_1}$ and ${Q_2}$ symbols. After interleaving, the symbols are mapped by the DMT mapping block. At the receiving side, each subcarrier is de-mapped by the corresponding QAM and recovered to binary bits after de-interleaving.

For TDHQ, the channel SNR is equalized and the input bitstream is mapped according to the equalized channel SNR. Then, precoding algorithms such as DFTS equalize the QAM symbols in a DMT symbol, and the SNR is equalized. At the receiver side after DMT de-mapping, the DMT symbol is decoded and recovered to binary bits by the corresponding QAM order.

For LC, the power coefficients and the QAM orders of each subcarrier are calculated. Then the input bitstream of each subcarrier is mapped into QAM symbols according to the corresponding QAM order and multiplied by the corresponding power coefficient. At the receiving side, the symbols on each subcarrier are de-mapped by the corresponding QAM order.

For FBL, each subcarrier needs to be assigned a different number of binary bits according to the estimated subcarrier SNR. Then the binary bits on each subcarrier are mapped into QAM symbols by the corresponding QAM order, which is derived from the subcarrier SNR and Eq. (8). After DMT de-mapping, the QAM symbols are de-mapped by the corresponding QAM orders of each subcarrier and recovered to binary bits.

The parameters of DMT are listed as follows: (1). The subcarrier number is 512, and half are utilized for the conjugating part of DMT. (2). Upsample time is 2. (3). Cycle prefix length is 16 points for each DMT symbol. (4). The overall symbol number in a frame is 600. (5). 10 symbols in a frame are utilized as training sequences.

4. Results and discussion

Hardware equalization is a powerful tool to expand the signal bandwidth and improve system performance. The expression of the received signal can be expressed as

(35)$$R(\omega ) = S(\omega )H(\omega ) + N(\omega ), $$

where $S(\omega )$ is the transmitting signal before pre-equalization, $H(\omega )$ is the channel frequency response, and $N(\omega )$ is the noise power frequency response. If we design a hardware pre-equalization circuit with a transfer function of ${H^{ - 1}}(\omega )$, the noise power at high frequency will be amplified and the signal power at low frequency is depressed profoundly. This overcompensation will instead cause a decrease in system performance.

As Fig. 6 shows, the spectrum and estimated SNR at the receiving side without a pre-equalizer, that with a pre-equalizer with the transfer function of $\sqrt {{H^{ - 1}}(\omega )} $, and that with a pre-equalizer with the transfer function of ${H^{ - 1}}(\omega )$ are investigated. The hardware pre-equalization circuit is a T-bridge resonant circuit [7,11], and the overall bandwidth of the pre-equalizer is 550 MHz. All the three schemes are measured at their optimal operation point of V_pp and bias current. The average SNR is measured by the procedure of channel SNR estimation in the first and second paragraph of Section. 3. As Fig. 6(a) shows, the system without a pre-equalizer has a deep fading channel response and the overall average SNR is only 9.96 dB, which can only support an average SE of 2.51 BPS transmission below 7% FEC threshold. With the $\sqrt {{H^{ - 1}}(\omega )} $ pre-equalizer, the overall average SNR increases to 21.15 dB, which corresponds to an average SE of 5.58 BPS. However, when the transfer function of the pre-equalizer is ${H^{ - 1}}(\omega )$, the overall average SNR is decreased to 16.64 dB, and the corresponding average SE is 4.32 BPS. Therefore, overcompensating the channel response to get a “flat” spectrum is not a proper compensation scheme. We choose the pre-equalizer with the transfer function of $\sqrt {{H^{ - 1}}(\omega )} $ for our following experiment. As the bit allocation or power allocation algorithms are processed based on the system SNR performance, it is evident that they cannot perform well without a pre-equalizer.

Fig. 6. System performance of pre-equalization: (a) Power and noise amplitude without pre-equalizer; (b) Power and noise amplitude with $\sqrt {{H^{ - 1}}(\omega )} $ pre-equalizer; (c) Power and noise amplitude with $ {{H^{ - 1}}(\omega )} $ pre-equalizer.

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The pre-equalization can be treated as another kind of “power allocation,” which decreases the low-frequency response and increases high-frequency response. Although the noise power is hard to measure, it can be estimated by the channel SNR and the measured power spectrum, which can be expressed by

(36)$${N_i} = {S_i} - \textrm{SN}{\textrm{R}_i}{\kern 1pt}, $$

where ${N_i}$ is the noise power amplitude on the i^th subcarrier and ${S_i}$ is the received signal power amplitude on the i^th subcarrier. All variables in Eq. (36) are in dB. The noise power amplitude is depicted as blue lines in Figs. 6(a)–6(c). It is easy to recognize that the noise power amplitude changes with different pre-equalization schemes in a large bandwidth such as 550 MHz. Therefore, we could not achieve the SNR performance we want.

To prove the feasibility of the algorithm presented in this study, we conducted a confirmatory study on the experimental setup shown in Fig. 5. We measured the optimal operation point of the UVLC system by a pre-equalizer with the transfer function $\sqrt {{H^{ - 1}}(\omega )} $. Figure 7 shows the contour of average SE below 7% FEC threshold. The optimal SE of 5.58 is measured at 130 mA bias current and 0.9 V V_pp. Therefore, we adjusted the operation point to 130 mA and 0.9 V to measure the highest data rate.

Fig. 7. Contour of SE performance at different V_pp and bias current.

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First, the system performance of 2DBA, FBL, LC, and TDHQ are measured at a relatively low bandwidth, at which ISI is relatively low. The spectrum of these four schemes is shown in Fig. 8. The bandwidth is adjusted to 300 MHz by changing the AWG sample rate, and the V_pp changes from 0.5 V to 1.2 V. Here we chose 0.9 V and 1.2 V as test points to measure the system spectrum. We can easily recognize that as 2DBA, FBL, and TDHQ distribute power uniformly, the spectrum of each is similar, whereas the spectrum of LC is a little different from those of 2DBA, FBL, and TDHQ, which shows that the power is not distributed uniformly on the overall subcarriers.

Fig. 8. Frequency response at 300 MHz: (a)–(d) 0.9 V frequency response of 2DBA, FBL, LC, and TDHQ; (e)–(h) 1.2 V frequency response of 2DBA, FBL, LC, and TDHQ.

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Figure 9 shows the differences in the SE of the four modulation schemes. Figures 9(a)–9(d) are the SE at 0.9 V V_pp, and Figs. 9(e)–9(f) are that at 1.2 V. The colour bar on the right side represents the SE of each subcarrier. The Y-axis is the symbol index in the time domain, and the X-axis is the subcarrier index in the frequency domain. To show the difference more intuitively between the SE of A and B at 0.9 V and 1.2 V, we did not interleave the symbols in Fig. 9, which is different from the actual experimental process. From Fig. 9, we can see that the SE at 0.9 V is much larger than that at 1.2 V. As the SNR is equalized, TDHQ has only two modulation formats for all the subcarriers, which is very different from the multiple modulation formats of 2DBA. For TDHQ, we sorted the SNR and achieved the maximum total number of bits at each operating point by dropping the worst subcarriers. Generally, we drop 1-2 subcarriers to achieve the best system performance. At the operation point of 130 mA, 0.9 V, and 550 MHz bandwidth, we have to drop 4 subcarriers to achieve the maximum number of bits in a frame. The 2DBA process performs hybrid modulation both in the time domain and the frequency domain, which is a two-dimensional hybrid modulation scheme.

Fig. 9. SE at 300 MHz and different V_pp. 0.9V: (a) FBL, (b) LC, (c) 2DBA, (d) TDHQ; 1.2 V: (e) FBL, (f) LC, (g) 2DBA, (h) TDHQ.

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The BER and data rate of 2DBA, FBL, LC, and TDHQ are shown in Fig. 10. As the E-tight LC algorithm cannot meet the FEC threshold, we chose to reduce the number of bits allocated to the LC algorithm one by one, and then test the BER performance of the system until the 7% HD-FEC error threshold was met. For the TDHQ algorithm, the performance is greatly deteriorated by the nonlinear distortion and ISI, as TDHQ can be treated as the single carrier signal. Therefore, the nonlinear equalizers are necessary to realize a better system performance for the TDHQ algorithm. We chose the least mean square equalizer with the Volterra series to equalize the received time domain signal after synchronization, which introduces much more computational complexity. The taps vary from 23 to 51, and increase with the V_pp and data rate. From Fig. 10(a), we can see that the BER performance of each of the four schemes is below the 7% FEC threshold. The data rate performance versus V_pp at 300 MHz is shown in Fig. 10(b). From 0.5 V to 0.9 V, the LC algorithm outperforms 2DBA as the system does not suffer from severe nonlinear distortion. However, when we adjust the system V_pp to 0.9 V and above, the nonlinear distortion begins to deteriorate the performance of the LC algorithm, and 2DBA begins to outperform LC. The highest data rate of 2.11 Gb/s at the bandwidth of 300 MHz is achieved at 0.9 V by 2DBA. As FBL is limited by its SE, the data rate of FBL is the lowest. For some operation points such as 0.7 V and 1.0 V, TDHQ shows the optimal performance among the four schemes. However, when we adjust the bandwidth to a higher level, the deep fading at high frequency introduces a lot of noise to the TDHQ system and deteriorates the system performance. Furthermore, the granularity of TDHQ SE is coarse (only eight levels between two adjacent QAM orders), which usually cannot meet the high speed demand. The received constellations of 2DBA are shown in Fig. 10(c).

Fig. 10. BER and data rate of FBL, 2DBA, LC, and TDHQ at 300 MHz: (a) BER performance versus V_pp; (b) Data rate performance versus V_pp.; (c) 2DBA received constellations at 0.9 V.

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The system performance at the bandwidth of 550 MHz is also investigated. The spectra of 2DBA, FBL, LC, and TDHQ at 550 MHz are shown in Fig. 11. From Fig. 11, we can know that the channel attenuates heavily when the frequency is higher than 500 MHz, which shows a poor frequency response. It can be easily recognized that the spectrum of LC is distorted by power allocation, which is different from those of 2DBA, FBL, and TDHQ.

Fig. 11. Frequency response at 550 MHz. (a)–(d): 0.9 V frequency response of 2DBA, FBL, LC, TDHQ; (e)–(h): 1.2 V frequency response of 2DBA, FBL, LC, TDHQ.

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The SE at 550 MHz is shown in Fig. 12. Similar to Fig. 9, we did not interleave the symbols in Fig. 12, which is different from the actual experimental process. The SE at 0.9 V is much greater than that at 1.2 V. As the increase in bandwidth introduces ISI and the fading of the high frequency signal becomes severe, the SE of the 550 MHz bandwidth is significantly less than that of 300 MHz.

Fig. 12. SE at 550 MHz and different V_pp. 0.9 V: (a) FBL, (b) LC, (c) 2DBA, (d) TDHQ; 1.2 V: (e) FBL, (f) LC, (g) 2DBA, (h) TDHQ.

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The SNR, BPS, and power allocation of 2DBA, FBL, LC, and TDHQ at 550 MHz and 0.9 V are shown in Fig. 13. To better prove that power allocation cannot achieve the preset SNR performance in a large nonlinear high-bandwidth VLC system, we add the performance of LC with E-tight condition (marked as the hollow blue circle and labelled as “LC (above 7% FEC)”), which is shown in Figs. 13(b1)–13(b3). We utilize the received QPSK symbols to determine the estimated SNR, and the measured SNR is measured by the received QAM symbols of FBL, LC, 2DBA, and TDHQ. The worst subcarriers have been discarded to realize the highest data rate. The measured SNR of 2DBA and FBL matches well with estimated SNR due to the uniform power distribution. However, as Fig. 13(b1) shows, the SNR performance of LC with E-tight shows a power mismatch, which is caused by over-allocation of bits, nonlinear distortion, and strong ISI in the system. Decreasing the overall allocated bits causes the measured SNR to match the estimated SNR again, as Fig. 13(c1) shows. The SNR performance of TDHQ shows that the SNR is equalized by precoding. The BPS performance is shown in Figs. 13(a2)–13(e2). None of the algorithms can achieve continuously adjustable SE on each subcarrier except for the 2DBA algorithm. As Figs. 13(a3)–13(e3) shows, the power allocation of 2DBA, FBL, and TDHQ for each subcarrier is uniform, whereas the power on each subcarrier of LC is different.

Fig. 13. SNR, BPS, and power allocation of (a1)-(a3): 2DBA, (b1)-(b3): LC (above 7% FEC threshold), (c1)-(c3): LC (below 7% FEC threshold), (d1)-(d3): FBL, and (e1)-(e3): TDHQ.

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The BER and data rate performance of 2DBA, FBL, LC, and TDHQ versus different V_pp at 550 MHz are shown in Fig. 14. The optimal data rate at different V_pp is measured, and the result is shown in Fig. 14(b). The 2DBA algorithm outperforms FBL, LC below 7% FEC threshold, and TDHQ under all the operating conditions. The biggest data rate gap of 130 Mb/s between 2DBA and LC below 7% FEC threshold occurs at 1.2 V. Figure 14(c) shows the constellations of 2DBA at 1.2 V, where the data rate is 2.96 Gb/s.

Fig. 14. BER and data rate of FBL, 2DBA, LC, and TDHQ at 550 MHz: (a) BER performance versus V_pp; (b) Data rate performance versus V_pp; (c) 2DBA received constellations at 1.2 V.

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Finally, we measured the BER and data rate versus different bandwidth at 0.9 V and 130 mA bias current to obtain the highest data rate of different schemes. The results are shown in Fig. 15. All the schemes perform under the 7% FEC threshold. The 2DBA always performs the best in the overall testing range. With the increase of bandwidth, the data transmission rate continues to increase until the bandwidth reaches 550 MHz, and the data rate gap between 2DBA and LC increases. The biggest data rate gap 110 Mb/s between 2DBA and LC is measured at 575 MHz bandwidth. The highest data rates of 2DBA, LC below 7% FEC threshold, FBL, and TDHQ are 3.24 Gb/s, 3.20 Gb/s, 3.07 Gb/s, and 3.00 Gb/s, respectively. As far as we know, 3.24 Gb/s is the highest data rate reported in a single blue LED–based UVLC system [37]. The received constellations of 3.24 Gb/s at 550 MHz are shown in Fig. 15(c). The net rate after removing the redundancy such as cycle prefix and training sequences is 3.14 Gbps.

Fig. 15. BER and data rate of FBL, 2DBA, LC, and TDHQ at 0.9 V: (a) BER performance versus bandwidth; (b) Data rate performance versus bandwidth; (c) 2DBA received constellations at 550 MHz.

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5. Conclusion

In this paper, we propose a novel pre-equalization assisted nonlinear hybrid modulation scheme named two-dimensional bit allocation (2DBA). Experimental results showed that when the UVLC system suffers from high nonlinear effect and ISI at high bandwidth and high driving voltage, under which conditions the noise strongly depends on the optical power, the conventional power allocation scheme is no longer applicable in a high speed UVLC system. Furthermore, the 2DBA can outperform the power allocation–assisted LC algorithm and precoding-assisted TDHQ in terms of achievable data rate. The highest data rate of 3.24 Gb/s of 2DBA is measured at 0.9 V and 550 MHz bandwidth after 1.2 m underwater transmission. As far as we know, this is the highest data rate achieved in a single blue LED based UVLC system. The optimal power allocation scheme for UVLC system needs to be further investigated in the future.

Funding

National Key Research and Development Program of China (2017YFB0403603); National Natural Science Foundation of China (61925104).

Disclosures

The authors declare no conflicts of interest.

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FEC threshold	QAM order (M)
FEC threshold	2	4	8	16	32	64	128	256
0.0038	5.52	8.53	12.29	15.19	18.21	21.12	23.95	26.91
0.02	3.24	6.25	10.15	12.71	15.74	18.43	21.21	24.01
0.05	1.31	4.32	8.39	10.52	13.56	15.94	18.64	21.19

Underwater visible light communication at 3.24 Gb/s using novel two-dimensional bit allocation

Abstract

1. Introduction

2. Principle

2.1. SNR and BER estimation

2.2. Principles of 2DBA, LC, FBL, and TDHQ

2.3. Nonlinear effect and ISI distortion on power allocation

3. Experimental setup

4. Results and discussion

5. Conclusion

Funding

Disclosures

References

Cited By

Figures (15)

Tables (1)

Equations (36)

Optics Express