Probabilistic Amplitude Shaping Research Articles

Probabilistic Amplitude Shaping for Continuous-Variable Quantum Key Distribution With Discrete Modulation Over a Wiretap Channel

Probabilistic Amplitude Shaping for Continuous-Variable Quantum Key Distribution With Discrete Modulation Over a Wiretap Channel

FPGA based modified multi-repeat distribution matcher for probabilistic amplitude shaping

The technique of probabilistic amplitude modulation, based on distribution matching, has garnered considerable attention in recent years as a means to enhance spectral efficiency and diminish the constellation energy of coded modulation. This paper introduces the implementation of Probabilistic Amplitude Modulation (PAS) using a Modified Multi-Repeat Distribution Matcher (MMRDM) on a Field Programmable Gate Array (FPGA). The Modified Multiple Repetition Distribution Matcher (MMRDM) is integrated into a 2×2 Multiple Input Multiple Output Orthogonal Frequency Division Multiplexing (MIMO-OFDM) system, realized through the Xilinx System Generator (XSG). Simple Zero-Forced and Minimum Mean Squared Error (MMSE) equalizers are applied to the receiver for signal detection across the MIMO channel. The system incorporates enhanced security through chaos-based scrambling with 16 and 64 Quadrature Amplitude Modulation (QAM). VHDL code files for this system are generated for the Xilinx Kintex-7 (xc7k325t-3fbg676) for hardware implementation. Performance evaluation includes an assessment of required storage capacity, complexity, and bit error rate (BER). Using Vivado 2017.4, the system is successfully routed with resource utilization, for example, 0.67% Block RAM (BRAM), 68.6% Look-Up Tables (LUT), 83% DSP 48s, and 1.5% registers for 64-QAM uniform modulation. Similarly, for 64-QAM 10 level (shaper output 60 bit) shaped modulation, the resource utilization is 0.67% BRAM, 68.8% LUT, 83% DSP 48s, and 1.6% registers on the specified device. Simulation results demonstrate an improvement in the net shaping gain of approximately (2-4 dB) at 1×10^(-4) for different equalizer cases compared to uniform QAM, along with a notable reduction in required storage capacity and computational complexity.

Probabilistic Amplitude Shaping and Nonlinearity Tolerance: Analysis and Sequence Selection Method

Probabilistic amplitude shaping (PAS) is a practical means to achieve a shaping gain in optical fiber communication. However, PAS and shaping in general also affect the signal-dependent generation of nonlinear interference. This provides an opportunity for nonlinearity mitigation through PAS, which is also referred to as a nonlinear shaping gain. In this paper, we introduce a linear lowpass filter model that relates transmitted symbol-energy sequences and nonlinear distortion experienced in an optical fiber channel. Based on this model, we conduct a nonlinearity analysis of PAS with respect to shaping blocklength and mapping strategy. Our model explains results and relationships found in literature and can be used as a design tool for PAS with improved nonlinearity tolerance. We use the model to introduce a new metric for PAS with sequence selection. We perform simulations of selection-based PAS with various amplitude shapers and mapping strategies to demonstrate the effectiveness of the new metric in different optical fiber system scenarios.

Triple-Convex Probabilistic Constellation Shaping PAM8 in IM/DD System

A novel triple-convex 8-ary pulse amplitude modulation (TC-PAM8) probabilistic constellation shaping (PCS) scheme with bipolar symmetric alphabet is proposed in the intensity modulation/direct detection (IM/DD) system. Maintaining the basic implementation structure of the probabilistic amplitude shaping (PAS), the target entropy of TC-PAM8 is adjusted by setting the one-degree-of-freedom shaping factor accordingly. The performance of the proposed scheme is verified by an O-band 20-GBaud experimental system over 20-km standard single-mode fiber (SSMF) transmission, where the entropy is set to 2.2, 2.5, and 2.8 bit/symbol. The experimental results show that the proposed TC-PAM8 exhibits a better transmission performance compared with Maxwell-Boltzmann PAM8 (MB-PAM8) and Reverse-Maxwell-Boltzmann PAM8 (Rev-MB-PAM8) shaping scheme. Specifically, TC-PAM8 achieves a maximum gain of up to 1 dB and 2.2 dB in receiver sensitivity compared with the MB-PAM8 and Rev-MB-PAM8, respectively when the entropy is 2.2 bit/symbol.

Rate-Compatible Protograph LDPC Codes for Source Coding in Joint Source—Channel Coded Modulation Systems

Using the source residual redundancy in a joint source and channel coded modulation (JSCCM) system to achieve a shaping gain has proven to be a good solution for probabilistic amplitude shaping. In a JSCCM system, rate-compatible source codes are essential for adapting to dynamically changing source probabilities. However, conventional rate-compatible source codes are not appropriate for JSCCM systems, since they may yield a loss in the shaping gain. Moreover, a code design that depends solely on the source decoding threshold appears to be ill suited, thus complicating the search for good source codes. In this paper, we propose a rate-compatible family of source codes for JSCCM systems based on an achievable system rate analysis and the source protograph extrinsic information transfer (PEXIT) algorithm. The proposed codes not only have good source decoding thresholds but also obtain shaping gains over the whole considered range of source probabilities. Numerical results show that the proposed rate-compatible family of source codes can significantly improve the bit-error-rate (BER) performance of the JSCCM system.

On the Nonlinear Shaping Gain With Probabilistic Shaping and Carrier Phase Recovery

The performance of different probabilistic amplitude shaping (PAS)techniques in the nonlinear regime is investigated, highlighting its dependence on the PAS block length and the interaction with carrier phase recovery (CPR). Different PAS implementations are considered, based on different distribution matching (DM) techniques—namely, sphere shaping, shell mapping with different number of shells, and constant composition DM—and amplitude-to-symbol maps. When CPR is not included, PAS with optimal block length provides a nonlinear shaping gain with respect to a linearly optimized PAS (with infinite block length); among the considered DM techniques, the largest gain is obtained with sphere shaping. On the other hand, the nonlinear shaping gain becomes smaller, or completely vanishes, when CPR is included, meaning that in this case all the considered implementations achieve a similar performance for a sufficiently long block length. Similar results are obtained in different link configurations ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$1\times 180$</tex-math></inline-formula> km, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$15\times 80$</tex-math></inline-formula> km, and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$27\times 80$</tex-math></inline-formula> km single-mode-fiber links), and also including laser phase noise, except when in-line dispersion compensation is used. Furthermore, we define a new metric, the nonlinear phase noise (NPN) metric, which is based on the frequency resolved logarithmic perturbation models and explains the interaction of CPR and PAS. We show that the NPN metric is highly correlated with the performance of the system. Our results suggest that, in general, the optimization of PAS in the nonlinear regime should always account for the presence of a CPR algorithm. In this case, the reduction of the rate loss (obtained by using sphere shaping and increasing the DM block length) turns out to be more important than the mitigation of the nonlinear phase noise (obtained by using constant-energy DMs and reducing the block length), the latter being already granted by the CPR algorithm.

Adaptive Bit-Labeling Design for Probabilistic Shaping Based on Residual Source Redundancy

By using the residual source redundancy to achieve the shaping gain, a joint source-channel coded modulation (JSCCM) system has been proposed as a new solution for probabilistic amplitude shaping (PAS). However, the source and channel codes in the JSCCM system should be designed specifically for a given source probability to ensure optimal PAS performance, which is undesirable for systems with dynamically changing source probabilities. In this paper, we propose a new shaping scheme by optimizing the bit-labeling of the JSCCM system. Instead of the conventional fixed labeling, the proposed bit-labelings are adaptively designed according to the source probability and the source code. Since it is simple to switch between different labelings according to the source probability and the source code, the proposed design can be considered as a promising low complexity alternative to obtain the shaping gain for sources with different probabilities. Numerical results show that the proposed bit-labelings can significantly improve the bit-error rate (BER) performance of the JSCCM system.

Probabilistic Shaping for Trellis-Coded Modulation With CRC-Aided List Decoding

This paper applies probabilistic amplitude shaping (PAS) to cyclic redundancy check (CRC)-aided tail-biting trellis-coded modulation (TCM). CRC-TCM-PAS produces practical codes for short block lengths on the additive white Gaussian noise (AWGN) channel. In the transmitter, equally likely message bits are encoded by a distribution matcher (DM) generating amplitude symbols with a desired distribution. A CRC is appended to the sequence of amplitude symbols, and this sequence is then encoded and modulated by TCM to produce real-valued channel input signals. This paper proves that the sign values produced by the TCM are asymptotically equally likely to be positive or negative. The CRC-TCM-PAS scheme can thus generate channel input symbols with a symmetric capacity-approaching probability mass function. The paper provides an analytical upper bound on the frame error rate of the CRC-TCM-PAS system over the AWGN channel. This FER upper bound is the objective function used for jointly optimizing the CRC and convolutional code. Additionally, this paper proposes a multi-composition DM, which is a collection of multiple constant-composition DMs. The optimized CRC-TCM-PAS systems achieve frame error rates below the random coding union (RCU) bound in AWGN and outperform the short-blocklength PAS systems with various other forward error correction codes studied in Coşkun et al. (2019).

Probabilistically Shaped PAM-8 Transmission With Simplified Volterra Equalizer

Intensity-modulation direct-detection (IM/DD) probabilistically shaped (PS) 8-level pulse-amplitude modulation (PAM) signal is promising for high-speed intra-data-center networks. However, the challenges of a PS-PAM-8 transmission system are complicated distribution matcher (DM) and nonlinear equalizer. In this work, we propose a simple PS-PAM-8 transmission system, where the probabilistic amplitude shaping (PAS) encoder consists of an energy-level-assigned (ELA)-based DM and a low-density parity-check (LDPC) encoder. In the ELA-based DM, a simple lookup table is designed to realize an approximate bilateral Maxwell-Boltzmann distribution by assigning energy levels to various symbols. Consequently, the DM encoded bits are used as the amplitude bits, and DM label bits and LDPC check bits are designated as sign bits to implement PAS encoding. In addition, a simplified Volterra equalizer (SVE) is introduced to eliminate the inter-symbol interference and improve the receiver sensitivity of PS-PAM-8 signal, where the SVE only includes the 2-order kernel to reduce its complexity. We experimentally demonstrated the proposed PS-PAM-8 scheme and verified its outperformance over uniform PAM-8 signal in the back-to-back and 2-km standard single mode fiber (SSMF) transmission. Compared to the conventional linear equalizer, the SVE contributes approximate 1-dB improvement of the receiver sensitivity in the PS-PAM-8 2-km SSMF transmission, for the error-free performance of post forward error correction bit error ratio.

Symbol-Wise Puncturing for HARQ Integration With Probabilistic Amplitude Shaping

In this letter, we propose a symbol-wise puncturing scheme to support hybrid automatic repeat request (HARQ) integrated probabilistic amplitude shaping (PAS). To prevent the probability distribution distortion caused by the traditional sequential puncturing and realize the promised gain of PAS, we perform symbol-wise puncturing on the label sequence of the shaped modulation symbols. Our simulation results indicate that the proposed puncturing scheme achieves a stable shaping gain across the signal-to-noise ratio of at least 0.6 dB compared with the uniform case under the same throughput, while the gain of sequential puncturing drops rapidly with retransmissions. Moreover, in each transmission, the proposed scheme is able to reduce the distribution distortion that achieves over 1.2 dB gain at a block error rate (BLER) of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\boldsymbol {10^{-3}}$ </tex-math></inline-formula> . In contrast, for sequential puncturing, the distribution is severely distorted and the BLER performance is even worse than that of the uniform case in retransmissions.

Dual Coding Concatenation for Burst-Error Correction in Probabilistic Amplitude Shaping

We propose the use of dual coding concatenation for mitigation of post-shaping burst errors in probabilistic amplitude shaping (PAS) architectures. The proposed dual coding concatenation for PAS is a hybrid integration of conventional reverse concatenation and forward concatenation, i.e., post-shaping forward error correction (FEC) layer and pre-shaping FEC layer, respectively. A low-complexity architecture based on parallel Bose–Chaudhuri–Hocquenghem (BCH) codes is introduced for the pre-shaping FEC layer. Proposed dual coding concatenation can relax bit error rate (BER) requirement after post-shaping soft-decision (SD) FEC codes by an order of magnitude, resulting in a gain of up to 0.25 dB depending on the complexity of post-shaping FEC. Also, combined shaping and coding performance was analyzed based on sphere shaping and the impact of shaping length on coding performance was demonstrated.

Rate-Adaptive Concatenated Multi-Level Coding With Novel Probabilistic Amplitude Shaping

This paper proposes a new probabilistic amplitude shaping (PAS) approach for concatenated two-level multi-level coding (MLC). The proposed system is based on a concatenated forward error correction (FEC) scheme where outer codes are serially concatenated with inner two-level MLC. This concatenated two-level MLC scheme has recently been shown to have a potential for achieving better performance-complexity trade-offs than the conventional bit-interleaved coded modulation (BICM). Meanwhile, PAS has recently been demonstrated to offer remarkable performance gains as well as rate adaptivity. However, the majority of existing works on PAS assume the use of the binary reflected Gray code as a bit-labeling, and its application to coded modulation schemes with other bit-labelings, such as two-level MLC, may not be straightforward. In this paper, we devise a bit-labeling scheme and propose a new PAS structure for an efficient integration of PAS with two-level MLC systems. More specifically, we propose to generate <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">signed</i> amplitude symbols with the distribution matcher (DM) for maximizing both coding and shaping gains achieved by two-level MLC and PAS, respectively, while the conventional PAS generates <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">unsigned</i> amplitude symbols. It is demonstrated by simulation results that, with 256QAM and inner polar codes, the proposed two-level MLC with PAS simultaneously offers 75% reduction in the number of required inner encoding and soft-decision (SD) decoding operations for given outer and inner FEC code lengths, and up to 0.3 dB performance gain over the conventional PAS scheme.

Real-Time FPGA Investigation of Interplay Between Probabilistic Shaping and Forward Error Correction

In this work, we implement a complete probabilistic amplitude shaping (PAS) architecture on a field-programmable gate array (FPGA) platform to study the interplay between probabilistic shaping (PS) and forward error correction (FEC). Due to the fully parallelized input–output interfaces based on look up table (LUT) and low computational complexity without high-precision multiplication, hierarchical distribution matching (HiDM) is chosen as the solution for real time probabilistic shaping. In terms of FEC, we select two kinds of the mainstream soft decision-forward error correction (SD-FEC) algorithms currently used in optical communication system, namely Open FEC (OFEC) and soft-decision quasi-cyclic low-density parity-check (SD-QC-LDPC) codes. Through FPGA experimental investigation, we studied the impact of probabilistic shaping on OFEC and LDPC, respectively, based on PS-16QAM under moderate shaping, and also the impact of probabilistic shaping on LDPC code based on PS-64QAM under weak/strong shaping. The FPGA experimental results show that if pre-FEC bit error rate (BER) is used as the predictor, moderate shaping induces no degradation on the OFEC performance, while strong shaping slightly degrades the error correction performance of LDPC. Nevertheless, there is no error floor when the output BER is around 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-15</sup> . However, if normalized generalized mutual information (NGMI) is selected as the predictor, the performance degradation of LDPC will become insignificant, which means pre-FEC BER may not a good predictor for LDPC in probabilistic shaping scenario. We also studied the impact of residual errors after FEC decoding on HiDM. The FPGA experimental results show that the increased BER after HiDM decoding is within 10 times compared to post-FEC BER.

List-Encoding CCDM: A Nonlinearity-Tolerant Shaper Aided by Energy Dispersion Index

Recently, a metric called energy dispersion index (EDI) was proposed to qualitatively indicate the nonlinear interference (NLI) induced by correlated symbols during optical transmission. In this paper, we propose a new shaper architecture to decrease the EDI of transmitted symbols and thus, increase the signal-to-noise ratio (SNR). We call this shaper the list-encoding constant-composition distribution matcher (L-CCDM). L-CCDM consists of an additional EDI selecting module, which is compatible with standard probabilistic amplitude shaping (PAS) architecture. Numerical results obtained from a multi-span multi-channel system show that when compared to standard CCDM with 256-ary quadrature amplitude modulation (256QAM), the proposed architecture offers an effective SNR gain of 0.35 dB, an achievable information rate gain of 0.22 b/4D-symbol, or equivalently an 8% reach extension.

A Low-Complexity Probabilistic Amplitude Shaping With Short Linear Block Codes

We propose a new probabilistic amplitude shaping (PAS) scheme based on short linear block codes. In the proposed system, the capacity-approaching signal distribution is generated in the process of decoding linear block codes for a given information bit sequence. We associate the design problem of shaping codes with the classical covering problem, suggesting the use of good covering codes for shaping. From simulation results, it is demonstrated that by selecting <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">perfect</i> binary codes as our shaping codes, the proposed scheme offers a shaping gain of around 0.3–1.0 dB. By comparing with the enumerative sphere shaping (ESS) of the same block length, we verify that the proposed scheme achieves significantly lower storage complexity and computational complexity at the receiver even with comparable block error rate performance.

Temporal Energy Analysis of Symbol Sequences for Fiber Nonlinear Interference Modelling via Energy Dispersion Index

The stationary statistical properties of independent, identically distributed (i.i.d.) input symbols provide insights on the induced nonlinear interference (NLI) during fiber transmission. For example, kurtosis is known to predict the modulation format-dependent NLI. These statistical properties can be used in the design of probabilistic amplitude shaping (PAS), which is a popular scheme that relies on an amplitude shaper for increasing spectral efficiencies of fiber-optic systems. One property of certain shapers used in PAS -- including constant-composition distribution matchers -- that is often overlooked is that a time-dependency between amplitudes is introduced. This dependency results in symbols that are non-i.i.d., which have time-varying statistical properties. Somewhat surprisingly, the effective signal-to-noise ratio (SNR) in PAS has been shown to increase when the shaping blocklength decreases. This blocklength dependency of SNR has been attributed to time-varying statistical properties of the symbol sequences, in particular, to variation of the symbol energies. In this paper, we investigate the temporal energy behavior of symbol sequences, and introduce a new metric called energy dispersion index (EDI). EDI captures the time-varying statistical properties of symbol energies. Numerical results show strong correlations between EDI and effective SNR, with absolute correlation coefficients above 99% for different transmission distances.

LDPC-coded 50-Gbaud IM/DD PAM-4 transmission employing probabilistic amplitude shaping based on prefix-free code distribution matching

We propose and experimentally demonstrate the intensity-modulation direct-detection (IM/DD) transmission employing low-density parity-check (LDPC) coded probabilistic shaping 4-level pulse amplitude modulation (PS-PAM4), assist with pairwise-distributed probabilistic amplitude shaping (PAS) structure based on prefix-free code distribution matching (PCDM). Unlike traditional PAS, the probability of adjacent symbols in the pairwise-distributed PAS architecture is consistent, which facilitates the filling of check bits generated by LDPC in IM/DD systems. On the other hand, PCDM has attracted increasing attention in recent years because of its asymptotic optimality similar to constant composition distribution matching (CCDM) in block length, and its high parallelism in hardware implementation. This is the first time that PCDM is applied in PAM4 signal transmission of PAS architecture in IM/DD system together with LDPC. Experiments of different code rates of LDPC and PCDM are performed. The experimental results show that the receiver sensitivity can be improved by more than 1 dB compared with the uniform-distributed signal transmission.

Bi-Directional OFDM Truncated PS-4096QAM Signals Transmission in a Full-Duplex MMW-RoF System at E-Band

A bi-directional photonics-aided millimeter-wave (MMW) transmission system at E-band is proposed in this article, which can be applied in future mobile networks. Only one pair of horn antennas are adopted in the experiment to support symmetrical wireless signal transmission. The center frequency of the downlink and uplink is set as 73.5 and 83.5 GHz, respectively, to avoid the interference effect. Moreover, the probabilistic shaping (PS) technique is used to adapt transmission rate and improve system capacity under a limited SNR. The time-domain Volterra series based equalizer effectively compensates the nonlinear distortion induced by the photoelectric devices. Based on the experimental results, the proposed full-duplex MMW-RoF system can support OFDM truncated PS (TPS)-4096QAM signal transmission, whose entropy is 10 bits/symbol/Hz. We set the probability of the outermost points to zero and the generated TPS-4096QAM signal is actually a PS-1936QAM one. The system simultaneously allows 7 Gbaud TPS-4096QAM signal downlink and uplink transmission over 2 m wireless distance, satisfying the NGMI threshold at 0.83. Moreover, the bi-directional 7 Gbaud PS-1024QAM signal transmission distance can be extended to 13.4 m when we adopt a higher NGMI threshold at 0.67. So the total rate of our proposed bi-directional transmission system is achieved to 100 Gbit/s. We also investigate the BER performance with different modulation formats. Such a system is a promising solution for future high-speed backhaul networks with multiple modulation format requirements.

Probabilistic Shaping for Protograph LDPC-Coded Modulation by Residual Source Redundancy

Probabilistic amplitude shaping (PAS) has proved to be a promising way to achieve the shaping gain for an additive white Gaussian noise channel with a distribution matcher (DM). However, the DM schemes may suffer a rate loss and increase both complexity and latency, when they are not suited for the input bit stream with redundancy. In this paper, a novel PAS strategy is proposed for a joint source and channel coded modulation system, where the residual source redundancy after source coding can be exploited to obtain both shaping and coding gains. It is shown that the residual source redundancy can be controlled by choosing the row weight distribution for source code, thus making the probability distribution of the modulated symbols be optimized under an appropriate interleaver design. To guarantee the error-floor performance, the source code is constrained by predicting the probability distribution of source bits within target finite-length frames. By jointly designing source and channel codes, the residual source redundancy can also be exploited to achieve the coding gain. Compared with the state-of-the-art code pairs, the proposed code pairs have better error-floor performance, and achieve higher coding and shaping gains without suffering from the rate-loss and the latency caused by DM.

Polar coded probabilistic shaping PAM8 based on many-to-one mapping for short-reach optical interconnection.

In this paper, a polar coded probabilistic shaping (PS) 8-ary pulse amplitude modulation (PAM8) based on many-to-one (MTO) mapping is investigated for short-reach optical interconnection. By ingeniously assigning parity bits to ambiguities positions, no extra PS redundancy and no complex distribution matcher are required in the scheme comparing to traditional probabilistic amplitude shaping (PAS). The noise distributions after different transmission distances are studied and an optimal clock recovery method for PS signal is proposed to degrade the impact of severe eye skew effect on BER performance. The experimental results show that up to 1.2 dB and 0.8 dB shaping gains are respectively achieved over back-to-back (BTB) and 2-km standard single mode fiber (SSMF) transmission. With the help of the proposed optimal clock recovery method in the PS scheme, the shaping gain is improved from 0.15 dB to 0.4 dB after 10-km transmission. Moreover, compared to low-density parity-check (LDPC) code, the polar coded PS-PAM8 can provide an additional coding gain of 2.2 dB with code length of 256, which proves the performance superiority of polar code in short code length. Therefore, the proposed polar coded PS-PAM8 with low complexity and satisfactory BER performance is believed to be an alternative solution for the cost-sensitive short-reach optical interconnection.