OFDM based visible light communication with probabilistic shaping

A visible light communication (VLC) system using probabilistically shaped orthogonal frequency multiplexed (OFDM) modulation is presented. The average symbol energy of probabilistically shaped OFDM modulation is investigated for different modulation orders of quadrature amplitude modulation (QAM) under different entropy scenarios. Furthermore, the symbol error performance of the probabilistically shaped OFDM system under additive white Gaussian noise (AWGN) channel condition is evaluated. The probabilistically shaped OFDM system outperforms the uniformly distributed in the symbol error ratio performance. For 256-QAM with an entropy of 7.84 bits, we have shown a 3 dB gain in signal-to-noise ratio (SNR) per symbol compared to uniform 256-QAM at a symbol error ratio of 10-3. This gain increases to 5.5 dB by decreasing the entropy to 6.80 bits.


INTRODUCTION
In recent years, visible light communication (VLC) based on light emitting diode (LED) attracts tremendous interests for high-speed indoor wireless communication [10]. The availability of vast and unregulated light spectrum which is free from electromagnetic interference makes it a plausible technology to complement radio frequency (RF) based communication systems. A fully connected Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org. LIOT'20, September 21, 2020, London, United Kingdom © 2020 Association for Computing Machinery. ACM ISBN 978-1-4503-8099-7/20/09. . . $15.00 https://doi.org/10.1145/3412449.3412548 wireless communication system of VLC offers high-speed, inherent security, and bidirectional multiuser communication [8]. Various forms of multicarrier modulation techniques such as orthogonal frequency division multiplexing (OFDM) have been applied to achieve higher spectral efficiency [15]. The combination of high order quadrature amplitude modulation (QAM) with OFDM has proven to provide high data rates of 2-Gb/s [9] and 3-Gb/s [14] using a single LED. These impressive VLC studies all use uniformly distributed symbols. But there exists a gap to the Shannon channel capacity with uniformly distributed symbols in a noise limited channel [13].
In optical fiber communication, probabilistic shaping has been considered as means of approaching the capacity limit. It has played a crucial role for several recent record-setting optical fiber communication experiments to enhance transmission reach and capacity [2,3,7,16,17]. In probabilistic shaping, constellation points are assigned with differing probabilities which will give the constellation Gaussian-like distribution. For that, a distribution matcher such as constant composition distribution matching (CCDM) is required to allocate independent input bits into output symbols with a desired probability distribution [12].
In this workshop paper, we present a DC-biased optical OFDM based VLC system with uniformly distributed and probabilistic shaping applied on a symbol-by-symbol basis. The impact of choosing different entropy levels for probabilistically shaped symbols on average symbol energy is examined. The error performance results for both cases of uniformly distributed and probabilistic shaping are presented.
The remainder of the paper is organized as follows. In Section 2, the concept of probabilistic shaping and achievable information rate are discussed. Section 3 presents the DC-biased optical OFDM system model while Section 4 discusses the comparative average symbol energy and error performance. Finally, concluding remarks are provided in Section 5.

THE CONCEPT OF PROBABILISTIC SHAPING
In regular square QAM constellation, all of the constellation points are distributed uniformly. In contrast, probabilistic shaping assigns symbols with differing probabilities and shapes the probability of occurrence of the constellation points. This is realized by transmitting low energy symbols more frequently than high energy symbols and hence saves energy. Fig. 1 graphically illustrates probabilistic shaped 64-QAM for four different entropy values. It can be seen that the lower the entropy values, the higher the probability for symbols in the most inside circle of the constellation to be chosen and hence the more frequently those symbols are used. An important element to map input information bits to probabilistically shaped symbols is a distribution matcher. Constant composition distribution matcher (CCDM), proposed in [12], provides practical, invertible, fixed-to-fixed (f2f) length distribution matcher and has drawn considerable interest in application of probabilistic shaping in optical fiber communication [1][2][3]5].

Achievable Information Rate Using Probabilistic Shaping
To approach Shannon's capacity limit of an additive white Gaussian noise (AWGN) channel, given the power constraint in the transmitter side, the channel must be fed by a Gaussian source [4]. This can be achieved by applying a probabilistic shaping to M-QAM symbols, which gives a Gaussian-like distribution over the input constellation. [3]. It has been shown that the best possible distribution with which to select constellation points is a Maxwell-Boltzmann distribution which gives ultimate shaping gain in any dimension [11]. In this work, we applied probabilistic shaping on a symbol-by-symbol basis and use normal distribution, with mean μ = 0 and variance σ 2 to generate M-QAM symbols. The resulting entropy is given as H = 1 2 log 2 2π σ 2 . For instance, for the probabilistic shaped 64-QAM shown in Fig. 1 (a), the entropy is chosen to be 5.88 bits and the σ 2 is determined from this value to be 203.067.
Consider an independent and identically distributed discrete channel input X = 1 , 2 , . . . , . This symbol-wise input X is the complex M-QAM symbols with unit energy, i.e. E[|X| 2 ] = 1. The modulation order denoted by takes on 2 , where is number of bits per symbol. We consider only the case where is even. Thus, -QAM has a square constellation and can be decomposed into its constituent one-dimensional √ -PAM constellation. This means that every QAM symbol can be considered as two consecutive PAM symbols that represent real and imaginary parts of the QAM symbol. The corresponding output of the channel is Y = 1 , 2 , . . . , .
The mutual information between channel input X and output Y is defined as; where Y ( ) = ∈X Y|X ( | ) ( ) is the marginal distribution of Y, with marginal distribution of X denoted as ( ) and E [·] denoting expectation. The channel can be considered as memoryless, fixed auxiliary, and two-dimensional circularly symmetric and can be described by the channel conditional probability density where σ 2 is the noise variance of the auxiliary channel.
To evaluate the mutual information using Eq. (1), a Monte Carlo simulations of input-output pairs ( , ) of the channel can be used as an estimation. This gives achievable information rate in bits per symbols, R SYM as a lower bound to the mutual information I (X; Y). That is: In Fig. 2, the simulation result of mutual information in bits per symbol of Eq. (3) is depicted. For probabilistic shaped symbols, the entropy used in the simulation for 4-QAM, 16-QAM, 64-QAM, and 256-QAM is 1.9, 3.8, 5.7, and 7.6 bits, respectively. The Shannon's capacity limit = log 2 (1 + SNR) is also shown as a reference. As shown in the figure, as the SNR increases, the mutual information of 4-QAM, 16-QAM, 64-QAM, and 256-QAM, for both uniformly distributed and probabilistic shaped, saturate to 2, 4, 6, and 8 bits/symbol, respectively. The gain in probabilistic shaping in each of these modulation orders can be seen in the regions where the SNR per symbol is lower than the SNR threshold where the mutual information starts to saturate. That is, in the low SNR regime, the mutual information of a probabilistic shaped symbols approaches the Shannon capacity limit. To illustrate, the inset in Fig. 2 shows that at 7 bits/symbol mutual information, 0.83 dB can be gained from probabilistic shaping over the uniformly distributed symbols while the gap to the Shannon limit with the probabilistically shaped symbols is 0.72 dB.

OPTICAL OFDM COMMUNICATION SYSTEM SIMULATION
A DC-biased optical OFDM based optical communication system block diagram is shown in Fig. 3. In this system, we start by generating random symbols. For comparison purpose we consider both uniformly distributed symbols and probabilistic shaped symbols.
(5) The time domain sample is then converted into serial and a cyclic prefix (CP) is added onto it to act as a guard band between successive OFDM frames and overcome intersymbol and intercarrier interference in dispersive (multipath propagation) optical wireless channel. The discrete sampled signal [ ] is then fed into digitalto-analog (DAC) converter to obtain a continuous time-domain signal ( ). To ensure unipolar time domain signal required for intensity modulation of the optical source (LED in this case), a DC bias is added. The condition ≥ |min( ( ))| must be met to ensure the signal, ( ) is unipolar signal.
The real unipolar time-domain signal ( ) is then used to drive the LED. We consider two types of white LEDs which are warm white LED and cool white LED. We model the response of the LEDs as a low-pass filter. It has been shown that warm white LEDs can be modelled as a first-order LPF and cool white LEDs as a third-order LPF [6].
At the receiver, the reverse operation to that of the transmitter side is done. The photodetector converts the optical signal back to electrical signal. The DC added in the transmitted is then removed before the time-domain continuous signal is sampled by analog-to-digital converter (ADC). Afterwards, the CP are removed and framed together in parallel. An FFT operation transforms the received signal ( ) into its frequency domain equivalent [ ] and the Hermitian symmetry component removed prior to -QAM de-mapping. The -QAM de-mapper perform QAM demodulation to estimate the transmitted symbols. From this, symbol error ratio of the system is evaluated by comparing the transmitted symbol with the received symbols.

RESULTS AND DISCUSSION
The DC-biased optical OFDM block diagram shown in Fig. 3 is simulated for both uniformly distributed and probabilistic shaped

Average Symbol Energy
To evaluate the gain in the average transmitted symbol energy of -QAM signal, the symbol energy of uniformly distributed symbols is compared with that of the probabilistically shaped symbols for the aforementioned four scenarios. The simulation results in Fig. 4 show that 256-QAM gives a larger gain in average symbol energy compared to other QAM constellations. This is due to the fact that lower energy symbols are used more frequently as illustrated in Fig. 1. Furthermore, the gain is higher in lower entropy scenarios. These results are expected as probabilistic shaping has more constellation shaping effect as the modulation order increases and in lower entropy, as well.

Symbol Error Ratio Performance
In this section, symbol error ratio comparison between uniformly distributed symbols and probabilistic shaped symbols is presented. In the simulations, 128 OFDM data-carrying subcarrier are used. The LED is modelled as a first-order LPF with a -3 dB cut-off frequency of 10 MHz and the sampling rate is taken as 20 MHz. The scenario with an entropy of 0.98 × log 2 is chosen for probabilistic shaped symbol. Fig. 5 shows the comparison in symbol error ratio between uniformly distribution and probabilistic shaping for different modulation orders. The theoretical symbol error ratio for uniformly distributed symbols in AWGN channel is also shown as a reference. For 4-QAM, as all four symbols are in a single constellation ring, there is no gain from using probabilistic shaping in symbols error ratio and hence it is not shown here. However, with higher modulation orders probabilistic shaping outperforms uniform distribution. This is shown in Fig. 5 (a), (b), and (c) for 16-QAM, 64-QAM, and 256-QAM, respectively. For achieving a symbol  error ratio of 10 −3 , 2.1 dB, 2.6 dB, 3 dB improvement is obtained in SNR per symbol for 16-QAM, 64-QAM, and 256-QAM respectively.
The error performance gain increases at lower entropy values for the probabilistically shaped symbols. This is shown in Fig. 6 for probabilistic shaped 256-QAM with entropy values of 6.80 bits and 7.84 bits. For the same symbol error ratio of 10 −3 , the lower entropy probabilistic shaping improves the gain by almost 2.5 dB. This energy saving makes probabilistic shaping very attractive for VLC in different scenarios, including underwater optical communication.

CONCLUSION
In this work, we have studied probabilistic shaping for DC-biased optical OFDM system. It has been shown that with probabilistic shaped 256-QAM and entropy of 6.8 bits, an average symbol energy reduction of up to 70% could be obtained relative to uniformly distributed 256-QAM. In terms of error rate, the probabilistic shaped -QAM outperforms the uniformly distributed -QAM in noise limited channel conditions. For probabilistically shaped 256-QAM with an entropy of 7.84 bits, we have shown 3 dB gain in SNR per symbol compared to uniform 256-QAM at a symbol error ratio of 10 −3 . This gain is increased to 5.5 dB by decreasing the entropy to 6.80 bits.