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Abstract
The probabilistic constellation shaping (PCS) technology has recently gained a great deal of attention for coherent optical communication systems since it allows us to approach the Shannon capacity limit by varying the symbol distribution adaptively to the signal-to-noise ratio (SNR). However, there is a lack of literature on how to apply this technology to intensity modulation (IM)/direct detection (DD) systems. In this paper, we propose and demonstrate an efficient way to apply the PCS technology for IM/DD systems. In the mapping of forward error correction-encoded bits onto the pulse amplitude modulation (PAM) symbols, we assign the uniformly distributed bits to the least significant bit of binary reflected Gray coding. Then, we have a pairwise distribution of symbol amplitude, where two adjacent symbols have the same probability. Although this distribution deviates from the optimum distribution (such as Maxwell-Boltzmann distribution), we show that the SNR penalty caused by this discrepancy is negligible. We evaluate the performance of the proposed scheme through simulation by measuring the achievable rate and frame error ratios after inverse distribution matcher. The results show that the proposed scheme provides a shaping gain larger than the time-division hybrid modulation. We also carry out the experimental demonstration of the proposed scheme using a 10-Gbaud PAM-8 signal. By using the proposed scheme, we improve the receiver sensitivity by 0.9 dB when compared with the uniformly distributed PAM-8 signal.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
High-level modulation formats, channel equalization using digital signal processing, and advanced error correction techniques have led to a significant capacity increase in fiber-optic communication systems for more than a decade. However, there still exists a gap of a few decibels in signal-to-noise ratio (SNR) between the capacity of modern fiber-optic communication system and the Shannon limit. The probabilistic constellation shaping (PCS) technology has recently attracted a great deal of attention in coherent optical communications as a means to close this gap. In this technology, low-amplitude symbols are transmitted more frequently than high-amplitude ones. Thus, by varying the kurtosis of a Maxwell-Boltzmann (MB) distribution of the symbol amplitude adaptively to the SNR, we can match the transmission rate very closely to the channel capacity with a fine granularity. Theoretically, the PCS technology gives us a maximum shaping gain of 1.53 dB for coherent systems in linear additive white Gaussian noise (AWGN) channel. The key component in the implementation of PCS technology is the constant composition distribution matcher (CCDM). This simple digital block converts the equiprobable binary information into the fixed-length probabilistically shaped (PS) symbols, without suffering from the error propagation problem observed in variable-length distribution matchers. The inverse distribution matcher (IDM) at the receiver performs the inverse mapping from the PS symbols to uniformly distributed (UD) bits. In this system, the forward error correction (FEC) should be incorporated to minimize the errors in the received symbols fed to the IDM. However, a technical challenge of amalgamating the PCS technology with FEC is that the symbol distribution (achieved by using CCDM) is not maintained after FEC encoding. This is because the parity bits tend to have a uniform distribution although they are generated using non-uniformly distributed bits [1]. An elegant solution to this problem was proposed in [2]. The PS bipolar symbol to be transmitted over the channel can be factorized into the unipolar symbol amplitude and its sign. The probability of each sign is 1/2. In the scheme proposed in [2], a systematic binary FEC having a rate of (k−1+γ)/k is employed, where k is an integer and γ is the fraction of sign used for data bits. Thus, the PS symbols expressed in nc(k−1) bits together with ncγ UD data bits, where nc is the channel use, are used to generate uniformly distributed nc(1−γ) parity bits. Then, each UD bit is transformed into the corresponding sign and multiplied by the PS symbol outputted from the CCDM. This simple scheme allows the amplitude of PS bipolar symbols (to be transmitted over the channel) to have the optimum MB distribution. Since its proposal in 2015, a lot of experimental works have been followed for coherent fiber-optic communication systems [3–6].
Direct detection (DD) is the simplest of optical detection schemes. Thus, DD receivers have been the receiver of choice for cost-sensitive applications. There have been several attempts to apply the PCS technology to intensity-modulation (IM)/DD systems [7–9]. In [7], a 10% increase in the net data rate was achieved by using PS 8-ary pulse amplitude modulation (PAM-8) signals. It was experimentally shown that the PCS technology improves the receiver sensitivity by nearly 1 dB when used for a band-limited IM/DD link [8]. However, the scheme proposed in [2] cannot be applied directly to IM/DD systems since the information should be on the unipolar intensity of light in DD system, rather than on the bipolar amplitude of light as in coherent-detection system. The previous works on the use of PCS technology for IM/DD systems sidestep this problem, and thus do not include the FEC encoding and decoding. For example, it was assumed that the amplitude distribution of IM signal followed the desired distribution (e.g., MB or exponential distributions) without taking into account the fact that the parity bits have a uniform distribution [7,8]. To the best of our knowledge, there is no report on how to transmit the uniformly distributed parity bits efficiently in IM/DD systems. It was only mentioned in [7] that the UD parity bits can be transmitted in a time-shared fashion such as using the time-division hybrid modulation (TDHM) [10], at the expense of performance degradation. It is conjectured that the similar method was utilized to transmit the parity bits in [9].
Hence, we present an efficient way to apply the PCS technology to IM/DD systems. In the PCS technology for coherent systems, an amplitude shift-keying (ASK) signal having a symmetric amplitude distribution is generated by assigning the sign to the most significant bit (MSB) and the unipolar amplitude of the signal to the rest of bits and then mapping the binary reflected Gray coded bits onto the ASK symbols. In this work, we propose a new mapping scheme where the parity bit is assigned to the least significant bit (LSB) and then the Gray coded bits are mapped onto the PAM symbols. As a result, two adjacent symbols have the same probability (due to the uniform distribution of the parity bit). Even though this leads to the deviation from the MB distribution, we show that the SNR penalty induced by this deviation is negligible. We evaluate the performance of our proposed scheme by measuring the achievable rate and frame-error rate (FER) after inverse distribution matcher for PAM-8 signals. We show through simulation that the proposed scheme provides the shaping gain of 1.5 dB with respect to UD PAM-8 signal. Also shown in the simulation is that the proposed scheme achieves a higher shaping gain than the TDHM scheme. We experimentally demonstrate the proposed scheme over a 10-Gbaud IM/DD link using a directly modulated laser (DML). The experimental results show that the proposed scheme improves the receiver sensitivity of PAM-8 signal by 0.9 dB, compared to the UD signal.
The remainder of the paper is organized as follows. In Section 2, we present the proposed FEC encoding scheme for PS IM/DD systems. The performance evaluations of the proposed scheme using simulation and experiment are given in Section 3 and 4, respectively. Finally, Section 5 concludes the paper.
2. Operation principle
It is known that the MB distribution maximizes the entropy in a linear AWGN channel under the constraint of signal power [2]. For IM/DD systems using the PCS technology, however, there is a mixture of distributions studied in the literature: MB and exponential distributions [7–9,11–13]. Thus, we utilize both distributions and evaluate their achievable rate.
2.1. Maxwell-Boltzmann and exponential distributions
Assuming that a PAM-M formatted signal is generated from the constellation set X = {0, 1, …, M-1}, the probability mass function (PMF) PX of MB distribution can be expressed as
where v is a rate parameter utilized to control the kurtosis of MB distribution.The PMF of exponential distribution is given by
The kurtosis of this distribution is also adjusted by varying ν.2.2. FEC encoding scheme for PS IM/DD system
Figure 1 shows the block diagram of the proposed PS IM/DD system. In this figure, we assume to utilize PAM-8 signal, but it can be applicable to PAM-M signals, where M is an even number. We also assume to utilize the MB distribution in this section. We utilize the CCDM to avoid the error propagation observed in variable-length distribution matchers. Firstly, we obtain the PS PAM-4 symbols, APAM-4 = [0,2,4,6], at the output of CCDM. The symbol amplitude follows the MB distribution, as shown in the inset. After labeling the symbols in the binary form, we send them to the systematic binary FEC encoder. For ease of explanation, we assume to utilize (k−1)/k-rate FEC, where k is an integer. For higher code rates, however, it is possible to utilize (k−1+γ)/k-rate FEC, where γ is the fraction of sign used for data bits, as shown in [2]. The PAM-4 symbols (expressed in bits) together with the parity bits are next mapped onto the PAM-8 symbols.
Fig. 1 Block diagram of the proposed PS IM/DD system. The insets show the amplitude probabilities of the symbols when PAM-8 signal is used.
Figure 2 shows the binary reflected Gray code (BRGC) mapping table for PAM-8 symbols after FEC encoding. The most significant difference between our proposed PCS scheme (for IM/DD system) and the conventional PCS scheme (for coherent system) is the mapping method. The first two bits in the mapping table are the PS PAM-4 symbols (labeled in bits). The LSB is assigned to the parity bit. We map these 3-bit data onto PAM-8 symbols such that only the LSB (i.e., the third bit for PAM-8 symbol) is different between two adjacent PAM-8 constellation points when grouped together from the lowest-amplitude PAM-8 symbol (i.e., ‘0’). For example, ‘000’ is mapped onto ‘0’ of PAM-8 symbol and ‘001’ onto ‘1’ symbol. We can see one-bit difference between ‘011’ and ‘010’ in Group B, which are mapped onto ‘2’ and ‘3’ of PAM-8 symbols, respectively. The BRGC mapping minimizes the number of bit errors when a symbol error occurs. In this way, we can generate PS PAM-8 symbols, APAM-8 = [0,1,2,3,4,5,6,7]. However, due to the uniform distribution of parity bits, the two symbols within a group have the same probability. This implies that the only even-indexed PAM-8 symbols follow the MB distribution, but the overall amplitude distribution of PAM-8 signal deviates slightly from the optimum MB distribution, as shown in the inset of Fig. 1. Hereafter this distribution is referred to as ‘pairwise MB distribution’. Nevertheless, the performance degradation caused by this discrepancy is negligible, which will be shown in Section 3.
2.3. Decoding scheme for PS IM/DD system
The detection process of the PS PAM-8 symbol is the inverse of the modulation and coding process. After the bit-metric decoding, we carry out the FEC decoding, inverse binary labeling, and finally inverse CCDM.
Following the framework in [15], we can write the achievable rate for probabilistic shaping as
where [⋅]+ = max(⋅,0), S is the shaping set, n is the output length of distribution matcher, q is decoding metric, X is the channel input, and Y is the channel output. Throughout our paper, a random variable and a vector are denoted by a capital letter and a bold-faced letter, respectively. The first term of (3) represents the number of information bits per symbol and the second term, U, represents the uncertainty. In the case of UD signals, the first term would be replaced with log2(M), where M is the modulation order. On the other hand, for the PS signals utilizing a distribution matcher having k input bits and n output symbols, the first term of (3) can be written as [15]Here, Rdm is the rate of distribution matcher. In the case of ideal PS signals where there is no rate loss at the distribution matcher, the first term of (3) can be expressed as the entropy of the transmitted channel input, H(PX) [16]. We can represent the transmitted symbol x in the binary form b = b1 … bm, where m is the number of bits. Then, for the bit-metric decoding, we utilize the decoding metric qbmd (b1…bm, y) = Π PBi|Y (bi |y), i = 1, 2,…, m, where PBi|Y (bi|y) is the conditional probability of bi given a received symbol y and Bi is the i-th bit of the transmitted symbol. Then, the uncertainty would be expressed as [15]where E[∙] is the expectation operation and H(Bi|Y) is the conditional entropy of Bi given a channel output Y. The log-likelihood ratio (LLR) for i-th bit, Li, is given bywhere Xk0 and Xk1 are the sets of PAM-M symbols whose k-th bit is either 0 or 1.3. Simulation and results
We evaluate the performance of our proposed scheme first through Monte Carlo simulation. For this purpose, we transmit PS, PS-TDHM, and UD PAM-8 signals and evaluate their achievable rates and bit-error ratios (BERs). To make a fair comparison between these signals, we set the information rate to be ~1.8 bit/symbol.
3.1. Simulation
Figure 3 shows the simulation setup. For the generation of PS PAM-8 signal, a truncated pseudo-random binary sequence (PRBS) (length = 33696 bits) is first fed to the CCDM. The distribution matcher converts the uniformly distributed bits into MB-distributed PAM-4 symbols. The block length of the CCDM’s output is 300. After binary labeling, the signal is FEC-encoded. We utilize an FEC coding specified in DVB-S2 [14]. It is composed of an outer BCH code and an inner low-density parity-check (LDPC) code. We employ a rate-3/4 LDPC code for PS PAM-8 signals. The code rate can be expressed as (k−1+γ)/k, where k = 3 and γ = 0.25. Thus, under this code rate, 25% of UD bits are assigned to additional data and BCH parity bits. In FEC-encoding process, we insert another truncated PRBS (length = 5208 bits) in the frame of FEC, as shown in the inset of Fig. 3, to have the information rate of 1.8 bit/symbol [≈3 × (33696 + 5208) /64800]. Then, the encoded binary data are mapped onto the PAM-8 symbols according to the BRGC table shown in Fig. 2. For example, the first and second bits in BRGC mapping come from the PS data in the frame (i.e., bit index between 1 and 43200) and the third bit is assigned to the UD data and parity (i.e., bit index between 43201 and 64800). It is worth mentioning that the ratio between PS and UD bits in the frame is exactly 2/1. Figure 4(a) shows the PMF of PS signal used in the simulation. It has the pairwise MB distribution, as explained in Section 2. After the PAM-8 mapping, we add one-dimensional AWGN to the signal. This is to emulate the receiver noise and to adjust the SNR. The signal is sampled and then the LLRs are calculated for bit-metric decoding. We then evaluate the achievable rate of the signal. We also measure the BER at the input of IDM (denoted as ‘post-FEC BER’) and the FER at the output of IDM (denoted as ‘FER after IDM’).
For comparison, we also generate PS-TDHM PAM-8 signal. The inset of Fig. 3 shows the frame structure of the PS-TDHM signal. The uniformly distributed parity bits are transmitted in a time-shared fashion. Thus, the consecutive three bits in the frame are mapped sequentially from the beginning of the frame onto the PAM-8 symbols. Figure 4(b) shows the PMF of the PS-TDHM PAM-8 signal. It clearly shows that, due to the parity bits uniformly mapped onto the PAM-8 symbols, the PMF deviates from the MB distribution as the symbol index increases. We show later that this incurs an SNR penalty when compared with our proposed scheme.
We generate UD PAM-8 signal to evaluate the shaping gain of PS and PS-TDHM signals. We utilize the 3/5-rate FEC code for the UD signal. In this case, a PRBS (length = 38688) is fed directly to the FEC encoder. The inset in Fig. 3 shows the frame structure of the UD PAM-8 signal after FEC encoding. The information rate of the UD signal is 1.8 bit/symbol (≈3 × 38688/64800). The bits in this frame are uniformly distributed. They are sequentially mapped onto PAM-8 symbols, just like PS-TDHM PAM-8 signal. Thus, the PAM-8 symbols are uniformly distributed, as shown in Fig. 4(c). The performance of UD signal is evaluated by BER after FEC decoding, which is denoted as ‘post-FEC BER’.
Following the framework in [15], we assume that the transmission channel is a memoryless AWGN channel. Then, the channel output Y can be expressed as Y = Δ X + Z, where Δ is the channel attenuation and Z is the Gaussian noise having zero mean and variance σ2. Then, the conditional probability PY|X (y|x) of (6) can be expressed as [15]
When we transmit ns PAM-8 (m = 3) symbols in our Monte Carlo simulation, we can calculate the LLR for k-th bit of l-th received symbol, Lk,l , by replacing PY|X (y|x) in (6) with (7). In our simulation, we estimate this LLR by using x and y. It should be mentioned that, the blind method can be used to estimate Δ, σ2, and PX from the received symbols in practical systems [17]. Finally, we utilize the following equation for the estimation of achievable rate [15]where bk,l is k-th bit of l-th transmitted symbol. In the case of PS signal, the input and output lengths of CCDM are 468 bits and 300 symbols, respectively. Thus, the first term of (8) becomes 2.56 bit/symbol for the PS signal. On the other hand, the CCDM input and output lengths for PS-TDHM signal are set to be 723 bits and 300 symbols, respectively. Since the PS-TDHM signal consists of 74.7% ( = 48408/64800) of PS and 25.3% [ = (64800-48408)/64800] of UD signals, the first term of (8) is equal to 2.56 ( = 0.747∙log2|S|/n + 0.253∙m). Since the first term of (8) represents the maximum achievable rate when there is no decoding error, the achievable rate of both PS and PS-TDHM signals would approach 2.56 bit/symbol asymptotically as the SNR increases.3.2. Simulation results
We first evaluate the achievable rate. Figure 5 shows the achievable rate versus the SNR for PAM-8 signals having uniform, MB, and exponential distributions. For MB and exponential distributions, we optimize the rate parameter (ν) for the achievable rate at each SNR. We first make a comparison between the MB and exponential distributions. The result shows that the MB distribution outperforms the exponential distribution slightly all over the SNR. For example, the SNR differences between the two distributions are 0.4 and 0.05 dB at SNRs of 5 and 25 dB, respectively. Also shown in this figure is the achievable rate of the signal having the pairwise MB distribution (proposed in our scheme). In this case, the rate parameter is optimized for the achievable rate at each SNR. Thus, it shows the maximum rate achievable by using the pairwise MB distribution. When we compare the achievable rate between the MB and pairwise MB distributions, we can see that the SNR penalty induced by using the pairwise MB distribution is negligible especially for SNRs higher than 5 dB. For example, the SNR penalty is measured to be 0.2 dB at an achievable rate of 1.8 bit/symbol. We show the achievable rate of the UD PAM-8 signal in this figure to measure the shaping gain of PS signals. The results show that the shaping gains of MB, exponential, and pairwise MB distributions are 2.0, 1.8, and 1.8 dB, respectively, at 1.8 bit/symbol. Interestingly, these shaping gains (in IM/DD system) are higher than 1.53 dB, which is the maximum shaping gain in linear coherent systems. To the best of our knowledge, there is no theoretical report about the maximum shaping gain in IM/DD systems. However, it is interesting to note that a 2-dB shaping gain was reported at an achievable rate of 1.7 bit/symbol for IM/DD system in [7]. The figure also includes the achievable rate of the PS PAM-8 signal encoded by using DVB-S2 shown in the inset of Fig. 3 (denoted as ‘PS’ in the figure). The shaping gain of this signal is 1.5 dB at the achievable rate of 1.8 bit/symbol. Finally, the achievable rate of the proposed scheme is compared with that of the PS-TDHM scheme. The results show that the PS-TDHM has a shaping gain of 0.5 dB, but it is outperformed by the proposed scheme by 1 dB. This should be attributed to the deviation of its PMF from the MB distribution for high-index symbols, as depicted in Fig. 4(b).
Figure 6 shows the BER performance. The BERs shown in this figure are all measured after FEC decoding. As a reference, we first measure the post-FEC BER of PS signal generated without using the CCDM. This is to exclude any performance degradation arising from the CCDM. The BER performance of this system is denoted as ‘ideal PS’ in the figure. The results show that the proposed scheme incurs an SNR penalty of 0.3 dB with respect to the ideal PS at a BER of 10−4. This comes from the achievable rate loss of 0.04 [ = H(PX) – log2|S|/n] arising from the limited length of DM’s output. Thus, it would decrease as we increase the block size of CCDM. The FER after the IDM requires additional 0.1-dB SNR at a BER of 10−4 since a sufficiently low symbol-error ratio is required at the input of IDM. Thus, the post-FEC BER and the FER after IDM would converge asymptotically as the BERs approach zero. The figure also shows the BER performance of UD PAM-8 signal. This signal requires an SNR of 17.8 dB to have a BER of 10−4. This implies that the proposed schemeprovides a shaping gain of 1.4 dB (with respect to the UD signal), which is similar to the shaping gain measured by using the achievable rate in Fig. 5. For comparison, we also shows the BER performance of PS-TDHM. This scheme offers a shaping gain of 0.5 dB, but underperforms the proposed scheme by 0.9 dB.
4. Experiment and results
We carry out experiment to demonstrate the proposed scheme. Figure 7 shows the experimental setup. First, both the PS and UD PAM-8 signals are generated offline (using the same procedure as the simulation) and ported to an arbitrary waveform generator. The 10-Gbaud electrical signal is amplified and then fed to the DML operating at 1.31 μm. The bias current of the laser diode is set to be 50 mA for the UD PAM-8 signal. However, it should be lowered to 38 mA for the PS signal to make the extinction ratio (ER) of the PS signal the same as that of UD signal. This is because the average value of PS PAM-8 signal is located close to symbol ‘2’, as shown in Fig. 7(b), not in the middle of symbols ‘3’ and ‘4’ as in UD PAM-8 signal of Fig. 7(a). As a result, the average output power of DML for the PS signal is reduced to 8.0 dBm (from 9.5 dBm for UD signal). It is possible to set the average output power of the PS signal to be the same as that of UD signal. In this case, however, we should replace the amplifier driving the DML to increase the peak-to-peak current of the PS signal. The ERs of both signals are set to be 12.0 dB. The 3-dB modulation bandwidth of DML is measured to be >15 GHz, which is broad enough to accommodate the 10-Gbaud signal. After transmission over 10-m standard single-mode fiber (SSMF), the signal is detected by using a PIN detector. The signal is captured by using a real-time scope (vertical resolution = 8 bit, sampling rate = 80 Gsample/s) for offline digital signal processing. Finally, we evaluate the BER performance by direct error counting. The number of bits used for the BER evaluation is 5.18 million bits. The net data rate is 18 Gb/s for both UD and PS signals in the experiment.
Figure 8 shows the measured BER as a function of the received optical power. For the UD signal, we achieve the error-free transmission when the pre-FEC BER is lower than 9.2 × 10−2. This is achieved at the received optical power higher than −17.1 dBm. On the other hand, the error-free transmission of the PS signal is achieved at the received optical power of −18.0 dBm after IDM. At this optical power, the pre-FEC is measured to be 5.4 × 10−2. It should be noted that we utilize the rate-3/5 and rate-3/4 LDPC codes for UD and PS signals, respectively, to make both signals have the same information rate. The results show that the receiver sensitivity gain of the proposed scheme is measured to be 0.9 dB.
5. Conclusion
We have proposed and demonstrated an efficient way to apply the probabilistic constellation shaping technology to IM/DD systems utilizing PAM signals. In this scheme, we assign one-bit parity to the least significant bit of the FEC-encoded binary data and map them onto PAM symbols. Due to the uniform distribution of the parity bits, the probability mass function of the symbol amplitude follows the pairwise Maxwell-Boltzmann distribution.
We first evaluate the performance of the proposed scheme using simulation. We measure the achievable rate as a function of SNR. The results show that the SNR penalty induced by using the proposed pairwise Maxwell-Boltzmann distribution is negligible when compared to the Maxwell-Boltzmann distribution. The performance is also evaluated by BER after FEC decoding and frame-error ratio after inverse distribution matcher. The results show that the proposed scheme outperforms the time-division hybrid modulation in terms of shaping gain. We demonstrate the proposed scheme experimentally using 10-Gbaud PAM-8 signals. By using the proposed scheme, we improve the receiver sensitivity (measured after FEC decoding and inverse distribution matcher) by 0.9 dB.
We believe that the proposed scheme provides an efficient and practical way to improve the performance of IM/DD system using probabilistically shaped PAM signals.
Funding
Institute for Information and Communications Technology Promotion (IITP) (2016-0-00083).
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