Open Access
Add to BibSonomy
Abstract
Ultraviolet C (UVC) micro light-emitting diode (LED) can achieve symbol communication rate up to 100Msps at distance 40 meters without transmitter-side lens to guarantee certain communication mobility. We consider what we believe to be a new scenario where high speed UV communciation is realized under unknown low-rate interference. The signal amplitude properties are characterized, and the interference intensity is categorized into three cases, namely weak, medium and high interference intensity. The achievable transmission rates for the three cases are derived, where the achievable transmission rate for medium interference intensity can approach those in weak interference intensity and strong interference intensity cases. We provide Gaussian approximation and related log-likelihood ratio (LLR) calculation, which are fed into the subsequent message-passing decoder. In the experiment, the data is transmitted with symbol rate 20 Msps under unknown interference with symbol rate 1 Msps, received by one photomultiplier tube (PMT). Experimental results show that the proposed interference symbol estimation approach shows negligibly higher bit error rate (BER) compared with those with perfect knowledge on the interference symbols.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
As traditional radio frequency (RF)-based wireless communication suffers the issues of electro-magnetic interference, optical wireless communication (OWC) has received extensive research interest due to its immunity [1–3]. To alleviate the alignment requirement in mobile communication, ultraviolet (UV) communication is adopted due to the atmospheric scattering [4]. In addition, due to the atmospheric absorption of solar radiation, a photon-counting-based detector with low dark noise can be adopted. High attenuation of UV spectrum in the atmosphere is helpful to improve the anti-jamming capability and convertness to interception.
Many researchers have studied the characteristics of UV communication in [5–10], where Monte-Carlo approach is adopted to simulate random scattering and strong attenuation. The channel characterization for single and double scattering events for non-line-of-sight (NLOS) communication are conducted in [5–7], while the channel bandwidth is investigated in [8]. Experimental measurements on path loss and pulse broadening in long-distance NLOS UV communications are reported in [9] and [10]. The Poisson channel capacity has been analyzed [11]. The capacity of continous-time poisson channel [12,13] and discrete-time poisson channel [14,15] has been studied, where the optimal distrbution for discrete-time Poisson channel has been researched in [14]. Recently, the capacity of MISO poisson channel [16] and MIMO poisson channel [17] has been explored.
With high-sensitivity photomultiplier tube (PMT) receiver, long-distance UV communication has been realized. In [18], 400 kbps real-time NLOS 266 nm laser-based UV communication system over 500 meters at frame-error rate (FER) below $10^{-5}$ is demonstrated. In [19], 1 Mbps real-time NLOS UV scattering communication with receiver diversity over 1 km is realized based on low-density parity-check code (LDPC) with FER 0.0132. However, due to bandwidth limitation on regular UV light-emitting diode (LED) or external modulator, the maximum transmission rate is typically in the order of Mega symbols per second.
On the other hand, as semiconductor techniques are developping rapidly in recent years, extensive works on UVC LED, espicially UVC micro-LED have been reported in [20]. Wide bandwith of UVC micro-LED implies significant potential for high-speed communication with symbol rate higher than 10Msps. In 2017, 71-Mbit/s data rate at wavelength of 294 nm with UV-B LED based on Orthogonal Frequency Division Multiplexing (OFDM) modulation at distance 8 cm is realized [21]. Later, a 262 nm high-bandwidth-III-nitride-micro-LED is fabricated, which can produce optical power of $196 \mu$W at transmition rate 1.1 Gbps with bit error rate (BER) lower than $3.8\times 10^{-3}$ at distance 0.3m [22]. Free-space UV communication with data rate of 2 Gbps via OFDM using a 276.8 nm micro-LED was achieved with pre-equalization over 3m, where the BER can reach $2.86 \times 10^{-3}$ [23]. Recently, UV-x, UV-B, and UV-C micro-LEDs are combined to achieve 10 Gbps with OFDM over 0.5 meters [24].
However, in real communication, the transmission distance is typically longer than those in the UVC micro-LED tests, e.g., longer than 10 meters. UV communication with distance 20 meters to 40 meters can be adopted for covert communication, where increasing the communication rate is desirable to decrease the communication duration and possibility of being detected, while NLOS feature and limited transmission range of UV communication can further decrease the possibility of being detected [6,25,26]. The challenge on higher-rate UV communication is the received signal characterization, which no longer lies in discrete-photon regime but in transition regime instead [27]. Recently, a UV communication system with transmission rate at 1 Gbps at distance of 100 meters at wavelength 285 nm is reported [28], via adopting a lens to concentrate UV light at the transmitter. In real application, if the communciation system mobility is more crucial, an LED is adopted without transmitter-side lens, to guarantee a wide transmission field-of-view angle.
To realize medium-rate communication using UVC micro-LED with a wide transmission field-of-view angle, in this work, we propose a novel signal detection approach based on the sampled signal amplitude. Due to weak received power under a wide transmission angle, a high-sensitivity PMT receiver is employed. To further realize high-sensitivity optical signal detection, an LED at center wavelength of 266nm in solar-blind optical spectrum is employed. Symbol rate with standalone communication up to 100 Mbps in distance between 20 meters and 40 meters can be achieved. We consider communication interference at a lower rate. The interference intensity can be categorized into three levels, weak, medium and high interference intensity. The achievable transmission rates for these three cases are derived, where the achievable transmission rate for medium interference intensity can approach the corresponding rates in weak interference intensity and strong interference intensity cases. Moreover, high-rate LDPC code, medium-rate-LDPC and Raptor code are adopted. The interference symbol and its position are estimated, based on which the calculation of the LLRs for desirable symbols is proposed. It is demonstrated that the proposed interference estimation approach shows negligibly higher BER compared with that under perfectly known interference symbol value and boundary.
The remainder of this paper is organized as follows. In Section 2, we introduce the system model on the UV system under the unknown interference and demonstrate the Gaussian characteristics of received signal by PMT. In Section 3, we investigate the UVC micro-LED communication under the interference of low-rate background communication, where an interference symbol estimation approach is proposed. Finally, Section 4 concludes the paper.
2. System model
2.1 UV communication system with co-transmission interference
We build an UVC-LED communication system with co-transmission interference, as shown in Fig. 1. The transmitter side consists of an arbitrary waveform generator (Keysight $33600A$ AWG), a RIGOL DP832x power supply and an UVC-LED array. The LED chip is LM1HQMC supplied by UVLEDTEK, whose current-voltage (I-V) plot, bandwidth and spectrum intensity are shown in Fig. 2. The peak intensity is at 266 nm. The UVC-LED array consists of four 266 nm LEDs, working at optical power 4 mW and current 20 mA. The LED is driven by the AWG via square waveform signal using a BiasTee to send OOK signals. Another high power LED is driven by a different AWG to interfere the channel.
The receiver part consists of a Hamamatsu PMT (R-7154 module), an oscilloscope and a desktop computer. The Hamamatsu PMT is adopted to convert the received optical signals to electrical signals, which is encapsulated via a 266 nm UV optical filter that can suppress the out-of-band background radiation. The PMT is adopted due to its high detection sensitivity for photon-level signal. An MSO-X 6004A oscilloscope is adopted to capture the signal waveform at sampling rate 200 MHz, which is then processed by the computer in an offline manner. Specifically, the receiver sequentially performs synchronization, channel estimation, symbol detection, and channel decoding. Since synchronization is standard, we primarily focus on channel estimation, symbol detection and channel decoding.
2.2 Received signal characterization
The summation of samples within a symbol duration is shown in Fig. 3. Kolmogorov-Smirnov (KS) test is also adopted to evaluate the accuracy of Gaussian approximation for the summation of samples with symbol rates from 1 Mbps to 100 Mbps at distances from 20 meters to 40 meters. As shown in Table 1, the summation of samples can be approximated by Gaussian distribution at 0.15 significance level, except for symbol rate 100 Msps at distance 40 meters. The p-value is a probability that measures the evidence against the null hypothesis. A smaller p-value provides stronger evidence against the null hypothesis [29].
Table 1. p-value in KS-test for Gaussian approximation of the sample summation
Let ${\cal L}_0$ and ${\cal L}_1$ denote the sets of symbols zero and one, respectively. Let $\mu _0$ and $\sigma ^2_0$ denote the mean and variance for symbol zero, respectively; and let $\mu _1$ and $\sigma ^2_1$ denote the mean and variance for symbol one, respectively. Let $S_k$ denote the summation of samples for symbol $k$. The following parameter estimation rule is adopted,
2.3 Standalone transmission results
We adopt LDPC [19] with belief propagation (BP)-based soft decoding, through message passing between variable nodes and check nodes. Assuming equal the prior probability for symbols one and zero, the initial LLR of each received symbol $j$ is calculated by
We then test the communciation system without interference to show the superiority of the Gaussian approximation over Poisson approximation. High rate LDPC (960,1152) and medium rate (960,1920) LDPC are both tested, as shown in Fig. 4. Obviously the Gaussian approximation is much more accurate than Poisson approximation.
3. SISO communication system with interference
3.1 Co-transmission interference characterization
Consider the scenario of high-rate UV short-range LED transmission interfered by long-range low-rate UV transmission. Such a scenario can be modeled by two LEDs sending OOK signals of different data rates towards one PMT, where the high-rate desirable signals are interfered by low-rate signals. The interference channel is illustrated in Fig. 5. Let $s^{(d)}$ and $s^{(i)}$ denote the desirable symbol and interference symbol, respectively. Let $p_d$ and $p_i$ denote the prior probabilities for $s^{(d)} = 1$ and $s^{(i)} = 1$, respectively. Note that there are four combinations on $(s^{(d)}, s^{(i)}) \in \{0, 1\}^2$, corresponding to four different superpositions of desirable and interference signals.
According to different strength levels of interference and desirable signals, the received signal can be classified into weak interference regime, medium interference regime and strong interference regime. Three different coding schemes, high-rate LDPC code, medium-rate LDPC code and Raptor code, are adopted for the above three interference regimes.
Let $\mu _{00}$, $\mu _{01}$, $\mu _{10}$ and $\mu _{11}$ denote the mean sample sum corresponding to four types of combinations for $(s^{(d)}, s^{(i)}) = (0,0)$, (0,1), (1,0) and (1,1), respectively; and let $\sigma ^2_{00}$, $\sigma ^2_{01}$, $\sigma ^2_{10}$ and $\sigma ^2_{11}$ denote the corresponding variances. These mean and variance values need to be estimated. Since the synchronization symbols for desirable signals are known, the corresponding interference symbols are estimated based on CL and INT states. Based on the start position estimate, let ${\cal I}_1$ and ${\cal I}_0$ denote the set of interference symbols labelled only by INT and CL, respectively. Note that some interference symbols are not labelled, which are not included into ${\cal I}_1$ or ${\cal I}_0$. Let ${\cal L}_{00}$ denote the set of desirable synchronization symbols zero with interference symbols in ${\cal I}_0$, ${\cal L}_{01}$ denote the set of desirable synchronization symbols zero with interference symbols in ${\cal I}_1$, ${\cal L}_{10}$ denote the set of desirable synchronization symbols one with interference symbols in ${\cal I}_0$, and ${\cal L}_{11}$ denote the set of desirable synchronization symbols one with interference symbols in ${\cal I}_1$.
First, the means and variances of four combinations can be estimated as follows,
The three cases, the corresponding coding schemes and the model estimation are elaborated as follows.
Case ${\cal C}_0$: The signal amplitude is significantly larger than the interference amplitude, where high-rate LDPC is adopted. Since the signals can easily be distinguished from interference, the interference is treated as background radiation without special processing. Since the four combinations can be mixed into two symbols 0 and 1, we adopt Gaussian mixture model (GMM) to describe the probability density.
Case ${\cal C}_1$: The signal amplitude is comparable to the interference amplitude, where medium-rate LDPC is adopted. When computing the LLR information for the desirable singal symbols, the interference is not negligible but does not block the transmission of desirable signals. When performing decoding, the interference state of each desirable symbol is estimated, which is then utilized to compute the LLR of each desirable symbol. The distribution density functions are given as follows,
Case ${\cal C}_2$: The interference amplitude is singificantly stronger than the signal amplitude, which blocks the transimission of desirable symbols. In such a case, Raptor code is adopted, where the desirable signal under interference symbol one is treated as erasure and the desirable signal under interference symbol zero is utilized for decoding. Since half of the received signal is randomly erasured, low rate Raptor code is adopted. The construction of Raptor code is elaborated according to the Appendix. The distribution density functions of interest are $f(x|s^{(d)}=0,s^{(i)}=0)$ and $f(x|s^{(d)}=1,s^{(i)}=0)$ as shown in Eq. (11) and Eq. (13), respectively.
Figure 6 shows the distributions of sample summation under case ${\cal C}_0$. The left subfigure shows the distributions corresponding to four combinations of desirable and interference symbols, and the right one shows the mixed distributions corresponding to desirable symbols $(s^{(d)})$ one and zero. It is seen that the mixed distributions can still be well separated.
Figure 7 shows the distributions corresponding to different combinations for desirable and interference symbols, for case ${\cal C}_1$ and case ${\cal C}_2$. It is seen that the mixed distributions for desirable symbols cannot be well separated, where the interference cannot be simply treated as noise. Note that under case ${\cal C}_2$, the distribution of low-rate interference signal moves to the rightside of the distribution of desirable signal, which implies strong interference region.
Table 2 shows the results of KS-test at three different cases, which proves the reliability of Gaussian approximation.
Table 2. p-value in KS-test for Gaussian approximation in three cases
3.2 Achievable transmission rates under interference
First, we characterize the probability density function of different combinations of $s^{(i)}$ and $s^{(d)}$. We define the following function,
Then, we have the following marginal function,The achievable rates corresponding to the three cases are given as follows.
Case ${\cal C}_0$: Since the four combinations are mixed into two symbols 0 and 1, the achievable rate is given by
whereCase ${\cal C}_1$: The achievable rate is given by
whereCase ${\cal C}_2$: The achievable rate is given by
whereGiven prior probability $p_i$ for the interference, prior probability $p_d$ for the desirable signal can be optimized to maximize the achievable rate for the three cases.
3.3 Transmission parameter estimation
During the transmission, we first need to estimate among cases ${\cal C}_0$, ${\cal C}_1$ and ${\cal C}_2$, which case the current channel belongs to. Such operation is performed via characterizing the interference based on known synchronization symbols for desirable signals. Since the value and boundary of interference symbols are unknown to the receiver side, the interference symbols need to be estimated.
3.3.1 Interference symbol boundary estimation
Assume that the interference symbol duration is $K$ times of the desirable symbol duration. To decode the desirable information, we need to detect whether the desirable symbol is interfered. Moreover, since the interference symbol duration is $K$ times of the desirable symbol duration, we also need to estimate the start position of the interference symbol duration. Assume that a desirable symbol is either fully interfered or fully not, neglecting the scenario of partially interfered.
The delay estimation of interference symbols can be performed in the following three steps:
Step 1 (Interference Symbol Positioning): Find all the zero symbols in the synchronization part and check if they have pulses. Zero symbols with photons are labeled as interfered (INT). If the distance between two INT symbols is shorter than $K$ symbols, the symbols between the two are also labeled as INT.
Step 2 (Clean Symbol Positioning): Zero symbols with no photons are labeled as clean (CL). If the distance between two CL symbols is less than $K$ symbols and does not cover INT, the symbols between the two are also labeled as CL.
Step 3 (Delay Estimation): We define “vague interference symbol” as interference symbol covering both CL and INT labels. Locate the start positon of interference based on minimizing the number of “vague interfefrence symbols”. Specifically, assuming known $K$, we find the start positioning of the interference symbol via examining all possible $K$ start positions, counting the corresponding number of “vague interference symbols”, and selecting the start position with the minimum number of “vague interference symbols”.
3.3.2 Misjudgment probability calculation
The single-threshold detection error probability is adopted to evaluate the interference intensity, which can be calculated as follows,
To differentiate cases ${\cal C}_0$, ${\cal C}_1$ and ${\cal C}_2$, we adopt the following threshold detection based on two thresholds $B_1$ and $B_2$.
The three transmission cases are detected by the receiver and then feedbacked to the transmitter, which chooses the corresponding transmission modes in the subsequent transmission.
3.4 LLR computation for BP decoding
The LLRs for desirable symbols are calculated based on the mean and variance estimate. Specifically, for cases ${\cal C}_1$ and ${\cal C}_2$, the interference state needs to be detected prior to calculating the LLRs. Such detection is performed in a blockwise manner corresponding to the interference symbol duration, where the summation of samples corresponding to each desirable symbol is denoted as $[y_1, y_2,\ldots, y_K]$. Let $\textbf {s} = [s_1, s_2,\ldots, s_K]$ denote the corresponding desirable signal symbols. We define the following likelihood functions corresponding to CL and INT states,
Generalized likelihood ratio test (GLRT) is adopted, given by the following function,
Assuming equal prior probability of symbols one and zero for the desirable symbols, the LLR can be calculated for the following three cases.
Case ${\cal C}_0$: Since the four combinations can be mixed into two symbols 0 and 1, we only need to calculate one type of LLR based on the estimate of mean and variance as follows,
Case ${\cal C}_1$: The LLR computation is based on the detection results on CL or INT. For CL and INT states, the LLRs are calculated in Eqs. (16) and (17), respectively,
Case ${\cal C}_2$: Since the LLRs for INT state are discarded, the LLR corresponding to CL state are given as follows,
The desirable symbols with INT state are treated as erasure and the corresponding LLRs are set to zero. Only the LLRs corresponding to CL state are adopted for Raptor decoding.
3.5 Experimental results
The system is set up in the corridor as shown in Fig. 8, with a UVC micro-LED and a regular UVC LED for the desirable signal and interference, respectively. The output waveforms of UVC micro-LED, interference LED, as well as the output waveforms of the PMT detector, are shown in Fig. 9. We change the positions and power of the two LEDs to create weak, medium, strong interference. The interference LED is supplied by Yingguang Innovation Technology and can only transmit data at 1 Mbps with power up to 100 mW. The signal LED is placed 35 meters from the receiver while the interfernce LED is placed at distances from 20 meters to 40 meters from the receiver. The power of signal LED is fixed at 4 mW and that of interfernce LED varies from 1 mW to 100 mW.
We calculate the achievable rates of all experiment data sets in three cases based on uniform distribution. In our experiment, we assume $p_d$ and $p_i$ are 0.5. In simulation, we change $p_d$ from 1 to 0, to find out the optimized capacity. Here are the results. From Fig. 10, we can see that the achievable rate of ${\cal C}_1$ approaches that of ${\cal C}_2$ as interfernce grows stronger. The achievable rates of ${\cal C}_1$ approach those of ${\cal C}_0$ as the interfernce gets weaker. The two thresholds between ${\cal C}_0$, ${\cal C}_1$ and ${\cal C}_2$ are also shown as two dashed lines in Fig. 10. Moreover, the optimized results are very close to experiment results, which means the achievable rate reaches the maximum when $p_d$ is nearly 0.5. We also simulate different interference situations to show the relationship between the achievable rates in three cases along with approaching performance as shown in Fig. 11. For each experimental data set, we generate a set of distributions via changing the means of symbols in ${\cal L}_{01}$ and ${\cal L}_{10}$ and keeping sum of the two means the same, to generate a series of distributions with interference intensities from 0.001 to 0.2. Note that the variances of the four distributions are not changed. We calculate the capacity results corresponding to different interference intensity levels, and obtain the average value among all experimental data sets correponding to each interference intensity level. The average capacity is shown in Fig. 11.
Fig. 9. Input signals of (a) UVC micro-LED, (b) interference LED and (c) output waveforms of the PMT.
We then investigate the decoding performance under three cases. In ${\cal C}_0$, the BER keeps under $1\times 10^{-3}$ before the interference intensity reaches 0.00875, as shown in Fig. 12(a). In Fig. 12(b), the BER is still zero until the interference intensity reaches 0.1. After that, the BER suddenly rises to 0.012. However, in Fig. 12(c), the BER still keeps at zero even at 0.2 interference intensity, since most interfered signal has been removed before decoding the desirable signal. Thus, we can set boundaries $B_1 = 0.00875$ and $B_2 = 0.1$ to classify three cases.
Fig. 12. lg(BER) versus interference intensity in cases ${\cal C}_0$, ${\cal C}_1$ and ${\cal C}_2$.
To further evaluate the performance of the proposed interference symbol estimation approach, we adopt two benchmarks with certain perfect knowledge, “Simulation-1” with known interference symbol boundary but unknown symbols, as well as “Simulation-2” with both known interference symbol boundary and perfect known interference symbols. The decoding BER based on the experimental measurement data is also shown in Fig. 12. It is seen that the BER of the proposed approach is slightly higher than those of “Simulation-1” and “Simulation-2”, but the degradation is negligible. The above results demonstrate the reliability of the interference symbol estimation approach proposed by this work.
We also calculate the delay estimation error by signal symbol length, as shown in Fig. 13. In every experiment, we only estimate the delay between signal and interference once. The error between real interference symbol boundary and the estimate is no more than one signal symbol, which proves the accuracy of the proposed estimation method.
4. Conclusion
In this paper, we consider the communication system where large-bandwidth UVC micro-LED transmission is interfered by low-rate transmission. Three cases of interference has been considered, under weak, medium and strong interference intensity. The achievable transmission rate for the three scenarios are derived, where the achievable transmission rate for medium interference intensity can approach the corresponding rates in weak interference intensity and strong interference intensity cases. Three diffenrent coding methods, high-rate LDPC, medium-LDPC and Raptor codes are adopted, respectively. Two interference intensity thresholds, 0.00875 and 0.1 are adopted to classify the three cases. Compared with traditional method based on Poisson distribution of photon number, the proposed signal detection and decoding approach shows significantly lower BER. Moreover, compared with the ideal cases with perfect interference symbol knowledge, the proposed interference symbol detection approach shows negligibly higher BER.
Appendix
Raptor code is a common fountain code consisting of LT (Luby transform) code as the inner code and a linear block code as the outer code, which is typically adopted for noise/noiseless channel with erasures [30–35]. The LT encoding process includes the following steps:
- (1) Randomly choose $d_n$ as the degree of the $n^{th}$ output symbol $C_n$ from a degree distribution function $\rho (d)$;
- (2) Uniformly choose $d_n$ symbols from the input symbols;
- (2) Set $C_n$ equal to the modulo two sum of the selected $d_n$ symbols.
The degree distribution function $\rho (d)$ determines the asymptotic performance of the LT and Raptor code. According to [36], we design Raptor code (4608,960) with rate-5/6 LDPC code and 1/4 LT code. The node degree distribution adopted in this work is shown as follows
At the receiver side, BP soft decoding algorithm is adopted on the joint Tanner graph of both LDPC and LT codes. The Raptor decoding process combines two BP decoding stages, LT decoding and LDPC decoding, starting with LT decoding as shown in Fig. 14.
The joint decoding consists of turbo process of both LDPC and LT decoders, alternating between $p$ iterations of LT decoder and $q$ iterations of LDPC decoder. At the end of each round, LDPC check is performed, and the decoding terminates when all LDPC parity checks are satisfied or the maximum number of iterations is reached. In this work, we set $p = q = 1$, and the maximum number of iterations to 30.
Funding
Key Research Program of Frontier Sciences of CAS (QYZDY-SSW-JSC003); National Natural Science Foundation of China (62171428).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. T. Cao, J. Song, and C. Pan, “Simplified closed-form single-scatter path loss model of non-line-of-sight ultraviolet communications in noncoplanar geometry,” IEEE J. Quantum Electron. 57(2), 1–9 (2021). [CrossRef]
2. T. Cao, T. Wu, C. Pan, and J. Song, “Single-collision-induced path loss model of reflection-assisted non-line-of-sight ultraviolet communications,” Opt. Express 30(9), 15227–15237 (2022). [CrossRef]
3. W. Lian, J. Liu, S. Cui, L. He, and Y. Zhang, “Design for network card module of USB optical modem based on LED visible light communication,” Chinese Automation Congress, 17453832 (2017).
4. L. R. Koller, Ultraviolet Radiation (John Wiley and Sons, 1965).
5. D. K. Borah, V. R. Mareddy, and D. G. Voelz, “Single and double scattering event analysis for ultraviolet communication channels,” Opt. Express 29(4), 5327–5342 (2021). [CrossRef]
6. H. Xiao, Y. Zuo, J. Wu, H. Guo, and J. Lin, “Non-line-of-sight ultraviolet single-scatter propagation model,” Opt. Express 19(18), 17864 (2011). [CrossRef]
7. Z. Shen, J. Ma, T. Shan, and P. Su, “Improved monte carlo integration models for ultraviolet communications,” in 20th International Conference on Communication Technology (IEEE, 2020), pp. 168–172.
8. H. Ding, G. Chen, A. K. Majumdar, B. M. Sadler, and Z. Xu, “Modeling of non-line-of-sight ultraviolet scattering channels for communication,” IEEE J. Select. Areas Commun. 27(9), 1535–1544 (2009). [CrossRef]
9. G. Chen, L. Liao, Z. Li, R. J. Drost, and B. M. Sadler, “Experimental and simulated evaluation of long distance nlos uv communication,” in 9th International Symposium on Communication Systems, Networks Digital Sign (IEEE, 2014), pp. 904.
10. L. Liao, R. J. Drost, Z. Li, T. Lang, B. M. Sadler, and G. Chen, “Long-distance non-line-of-sight ultraviolet communication channel analysis: experimentation and modelling,” IET Optoelectron 9(5), 223–231 (2015). [CrossRef]
11. M. R. Frey, “Information capacity of the Poisson channel,” IEEE Trans. Inf. Theory 37(2), 244–256 (1991). [CrossRef]
12. M. Davis, “Capacity and cutoff rate for Poisson-type channels,” IEEE Trans. Inf. Theory 26(6), 710–715 (1980). [CrossRef]
13. A. D. Wyner, “Capacity and error exponent for the direct detection photon channel. II,” IEEE Trans. Inf. Theory 34(6), 1462–1471 (1988). [CrossRef]
14. J. Cao, S. Hranilovic, and J. Chen, “Capacity-achieving distributions for the discrete-time poisson channel—part I: General properties and numerical techniques,” IEEE Trans. Commun. 62(1), 194–202 (2014). [CrossRef]
15. A. Lapidoth and S. M. Moser, “On the capacity of the discrete-time Poisson channel,” IEEE Trans. Inf. Theory 55(1), 303–322 (2009). [CrossRef]
16. S. M. Haas and J. H. Shapiro, “Capacity of wireless optical communications,” IEEE J. Select. Areas Commun. 21(8), 1346–1357 (2003). [CrossRef]
17. K. Chakraborty, S. Dey, and M. Franceschetti, “Outage capacity of MIMO Poisson fading channels,” IEEE Trans. Inf. Theory 54(11), 4887–4907 (2008). [CrossRef]
18. Z. Xu and M. Sadler, “Ultraviolet communications: potential and state-of-the-art,” IEEE Commun. Mag. 46(5), 67–73 (2008). [CrossRef]
19. G. Wang, K. Wang, C. Gong, and Z. Xu, “1 Mbps real-time NLOS UV scattering communication system with receiver diversity over 1 km,” IEEE Photonics J. 10(2), 1–13 (2018). [CrossRef]
20. H. Yu, M. H. Memon, and H. Jia, “Deep-ultraviolet LEDs incorporated with SiO2-based microcavities toward high-speed ultraviolet light communication,” Adv. Opt. Mater. 17(38), 2195 (2022). [CrossRef]
21. X. Sun, Z. Zhang, and A. Chaaban, “71-Mbit/s ultraviolet-B LED communication link based on 8-QAM-OFDM modulation,” Opt. Express 25(19), 23267–23274 (2017). [CrossRef]
22. X. He, E. Xie, and M. S Islim, “1 Gbps free-space deep-ultraviolet communications based on III-nitride micro-LEDs emitting at 262 nm,” Photonics Res. 7(7), B41–B47 (2019). [CrossRef]
23. S. Zhu, P. Qiu, and Z. Qian, “2 Gbps free-space ultraviolet-C communication based on a high-bandwidth micro-LED achieved with pre-equalization,” Opt. Lett. 46(9), 2147–2150 (2021). [CrossRef]
24. D. Maclure, J. Mckendry, L. Sun, M. Islim, and E. Xie, “10 Gbps wavelength division multiplexing using UV-x, UV-B, and UV-C micro-LEDs,” Photonics Res. 10(2), 516–523 (2022). [CrossRef]
25. M. Luettgen, J. Shapiro, and D. Reilly, “Non-line-of-sight single-scatter propagation model,” J. Opt. Soc. Am. 8(12), 1964–1972 (1991). [CrossRef]
26. Y. Hongwei, C. Shengli, and J. Honghui, “Non-line-of-sight multiscatter propagation model,” J. Opt. Soc. Am. 20(9), 2466–2469 (2009). [CrossRef]
27. X. Liu, C. Gong, S. Li, and Z. Xu, “Signal characterization and receiver design for visible light communication under weak illuminance,” IEEE Commun. Lett. 20(7), 1349–1352 (2016). [CrossRef]
28. D. M. Maclure, C. Chen, and E. Xie, “Hundred-meter Gb/s deep ultraviolet wireless communications using AlGaN micro-LEDs,” Opt. Express 30(26), 46811–46821 (2022). [CrossRef]
29. G. Marsaglia, W. W. Tsang, and J. Wang, “Evaluating Kolmogorov’s distribution,” Journal of Statistical Software 46(18), 1–4 (2003). [CrossRef]
30. A. Shokrollahi, “Raptor codes,” IEEE Trans. Inf. Theory 52(6), 2551–2567 (2006). [CrossRef]
31. M. Shirvanimoghaddam and S. Johnson, “Approaching the capacity of AWGN channels using multi-layer raptor codes and superposition modulation,” International Symposium on Information Theory, 16228642 (IEEE, 2016).
32. B. Sivasubramanian and H. Leib, “Fixed-rate Raptor codes over rician fading channels,” IEEE Trans. Veh. Technol. 57(6), 3905–3911 (2008). [CrossRef]
33. K. Yen, Y. Liao, and C. Chen, “Modified robust soliton distribution (MRSD) with improved ripple size for LT codes,” IEEE Commun. Lett. 17(5), 976–979 (2013). [CrossRef]
34. J. Chen, A. Dholakia, E. Eleftheriou, M. P. C. Fossorier, and X.-Y. Hu, “Reduced-complexity decoding of LDPC codes,” IEEE Trans. Commun. 53(8), 1288–1299 (2005). [CrossRef]
35. K. Turk and P. Fan, “Adaptive demodulation using rateless codes based on maximum a posteriori probability,” IEEE Commun. Lett. 16(8), 1284–1287 (2012). [CrossRef]
36. Y. Wang, N. Wu, and Z. Xu, “Study of Raptor codes for indoor mobile VLC channels,” Globecom Workshops 18472288 (IEEE, 2018).
