post-FEC Bit Error Rate Research Articles

Post-FEC performance evaluation of optical SDM systems with mode-dependent loss

This work is focused on the bit error rate (BER) performance of spatial division multiplexing (SDM) systems over an optical channel with mode-dependent loss or gain (collectively referred to in this paper as MDL). When the latter is non-negligible, the BER has a random nature that introduces the outage probability as an important performance metric for the system design and also impacts on the selection of a forward-error correction (FEC) scheme. In MDL-impaired SDM systems, the pre-FEC BER is a random variable whose probability density function (PDF) and coding gain depend on the signal-to-noise ratio (SNR) at the receiver input. Hence, the common and simple approach of adding a coding gain factor to the pre-FEC BER to obtain the post-FEC BER is not adequate, and numerical simulations are needed. In this paper, we simulate and analyze the performance in terms of post-FEC BER for two proposals of applying low-density parity check (LDPC) FEC encoder/decoder in a SDM system MDL-impaired and an optimal linear multiple input multiple output (MIMO) receiver. In the first one, the LDPC is applied independently to each mode, and in the second one, the LDPC is applied among all SDM modes. Simulation results indicate that the first proposal outperforms the second. Simplifications in the log-likelihood ratio (LLR) computations have also been considered.

Introducing 4D Geometric Shell Shaping for Mitigating Nonlinear Interference Noise

Four dimensional geometric shell shaping (4D-GSS) is introduced as an approach for closing the nonlinearity-caused shaping gap. This format is designed at the spectral efficiency of 8 b/4D-sym and is compared against polarization-multiplexed 16QAM (PM-16QAM) and probabilistically shaped PM-16QAM (PS-PM-16QAM) in a 400ZR-compatible transmission setup with high amount of nonlinearities. Reach increase and nonlinearity tolerance are evaluated in terms of achievable information rates and post-FEC bit-error rate. Numerical simulations for a single-span, single-channel show that 4D-GSS achieves increased nonlinear tolerance and reach increase against PM-16QAM and PS-PM-16QAM when optimized for bit-metric decoding ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\text{R}_\text{BMD}$</tex-math></inline-formula> ). In terms of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\text{R}_\text{BMD}$</tex-math></inline-formula> , gains are small with a reach increase of 1.7% compared to PM-16QAM. When optimizing for mutual information, a larger reach increase of 3% is achieved compared to PM-16QAM. Moreover, the introduced GSS scheme provides a scalable framework for designing well-structured 4D modulation formats with low complexity.

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.

Design and Performance Analysis of Probabilistically Shaped QAM Signals for Coherent FSO Systems with Gamma-Gamma Turbulence Channels

Probabilistic shaping (PS) is a promising technique to approach the Shannon limit. In this paper, we design a practical coded modulation scheme based on PS to improve the capacity of coherent free-space optical (FSO) links with quadrature amplitude modulation (QAM), where the fading channel follows the Gamma-Gamma distribution. The aim of this paper is to optimize the probability mass function (PMF) of the QAM signal points to achieve the maximum channel capacity. Due to the complexity of the objective function, the heuristic algorithm was employed to solve the optimization problem. To the best of the authors’ knowledge, the closed-form pairwise error probability (PEP) is first derived with the non-uniform signals under the turbulence channel. In addition, we measure the average symbol error rate (SER) and post-FEC bit error rate (BER) by the Monte Carlo simulation method. The numerical simulation results of both capacity and BER show that the proposed PS scheme is better than the uniform distribution. The post-FEC BER results show that the proposed PS scheme provides significant gains compared with the uniform scheme.

Variable-Rate VLSI Architecture for 400-Gb/s Hard-Decision Product Decoder

Variable-rate transceivers, which adapt to the conditions, will be central to energy-efficient communication. However, fiber-optic communication systems with high bit-rate requirements make design of flexible transceivers challenging, since additional circuits needed to orchestrate the flexibility will increase area and degrade speed. We propose a variable-rate VLSI architecture of a forward error correction (FEC) decoder based on hard-decision product codes. Variable shortening of component codes provides a mechanism by which code rate can be varied, the number of iterations offers a knob to control the coding gain, while a key-equation solver module that can swap between error-locator polynomial coefficients provides a means to change error correction capability. Our evaluations based on 28-nm netlists show that a variable-rate decoder implementation can offer a net coding gain (NCG) range of 9.96-10.38dB at a post-FEC bit-error rate of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-15</sup> . The decoder achieves throughputs in excess of 400Gb/s, latencies below 53ns, and energy efficiencies of 1.14pJ/bit or less. While the area of the variable-rate decoder is 31% larger than a decoder with a fixed rate, the power dissipation is a mere 5% higher. The variable error correction capability feature increases the NCG range further, to above 10.5dB, but at a significant area cost.

Post-FEC BER Benchmarking for Bit-Interleaved Coded Modulation With Probabilistic Shaping

Accurate performance benchmarking after forward error correction (FEC) decoding is essential for system design in optical fiber communications. Generalized mutual information (GMI) has been shown to be successful at benchmarking the bit-error rate (BER) after FEC decoding (post-FEC BER) for systems with soft-decision (SD) FEC without probabilistic shaping (PS). However, GMI is not relevant to benchmark post-FEC BER for systems with SD-FEC and PS. For such systems, normalized GMI (NGMI), asymmetric information (ASI), and achievable FEC rate have been proposed instead. They are good at benchmarking post-FEC BER or to give an FEC limit in bit-interleaved coded modulation (BICM) with PS, but their relation has not been clearly explained so far. In this paper, we define generalized L-values under mismatched decoding, which are connected to the GMI and ASI. We then show that NGMI, ASI, and achievable FEC rate are theoretically equal under matched decoding but not under mismatched decoding. We also examine BER before FEC decoding (pre-FEC BER) and ASI over Gaussian and nonlinear fiber-optic channels with approximately matched decoding. ASI always shows better correlation with post-FEC BER than pre-FEC BER for BICM with PS. On the other hand, post-FEC BER can differ at a given ASI when we change the bit mapping, which describes how each bit in a codeword is assigned to a bit tributary.

Demonstration of a 2.34 Gbit/s Real-Time Single Silicon-Substrate Blue LED-Based Underwater VLC System

We develop a real-time discrete multi-tone (DMT) transceiver based on field programmable gate array (FPGA) chips for a single chip silicon-substrate light-emitting diode (LED) based underwater visible light communication (UVLC). On-chip resource usages are analyzed and discussed. To improve bit error rate (BER) performance, a novel channel estimation technique utilizing hybrid inter-symbol frequency-averaging (Inter-SFA) and intra-symbol frequency-averaging (Intra-SFA) is proposed and investigated. The real-time DMT transceiver is experimentally verified in a silicon substrate blue LED-based UVLC system with a 1.2 m underwater link. By using the enhanced channel estimation, a gross bit rate of 2.34 Gbit/s real-time DMT signal over 1.2 m underwater transmission can be achieved with the BER of 3.5 × 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−3</sup> . What's more, multiple-symbol interleaved Reed-Solomon (RS) codes are employed to further improve BER performance. The real-time measured post-FEC BER of the DMT-UVLC with multiple-symbol interleaved RS (255, 191) codes can be improved by more than six orders of magnitude. As a result, error-free (less than 1 × 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−9</sup> ) transmission is observed in our real-time experiment. Furthermore, 1.485 Gbit/s 720p high-definition video underwater transmission is successfully demonstrated. To the best of our knowledge, it is the first time to demonstrate a real-time LED-based UVLC system with DMT modulation beyond Gbit/s.

Hierarchical Distribution Matching for Probabilistically Shaped Coded Modulation

The implementation difficulties of combining distribution matching (DM) and dematching (invDM) for probabilistic shaping (PS) with soft-decision forward error correction (FEC) coding can be relaxed by reverse concatenation, for which the FEC coding and decoding lies inside the shaping algorithms. PS can seemingly achieve performance close to the Shannon limit, although there are practical implementation challenges that need to be carefully addressed. We propose a hierarchical DM (HiDM) scheme, having fully parallelized input/output interfaces and a pipelined architecture that can efficiently perform the DM/invDM without the complex operations of previously proposed methods such as constant composition DM (CCDM). Furthermore, HiDM can operate at a significantly larger post-FEC bit error rate (BER) for the same post-invDM BER performance, which facilitates simulations. These benefits come at the cost of a slightly larger rate loss and required signal-to-noise ratio at a given post-FEC BER.

FPGA-Based Rate-Compatible LDPC Codes for the Next Generation of Optical Transmission Systems

In this paper, we propose a rate-compatible forward error-correcting (FEC) scheme based on low-density-parity check (LDPC) codes together with its software reconfigurable unified field-programmable gate array (FPGA) architecture. By FPGA emulation, we demonstrate that the proposed class of rate-compatible LDPC codes based on puncturing and generalized LDPC coding with an overhead from 25% to 46% provides a coding gain ranging from 12.67 to 13.8 dB at a post-FEC bit-error rate (BER) of 10−15. As a result, the proposed rate-compatible codes represent one of the strong FEC candidates of soft-decision FEC for both short-haul and long-haul optical transmission systems.

Low-complexity BCH codes with optimized interleavers for DQPSK systems with laser phase noise

The presence of high phase noise in addition to additive white Gaussian noise in coherent optical systems affects the performance of forward error correction (FEC) schemes. In this paper, we propose a simple scheme for such systems, using block interleavers and binary Bose---Chaudhuri---Hocquenghem (BCH) codes. The block interleavers are specifically optimized for differential quadrature phase shift keying modulation. We propose a method for selecting BCH codes that, together with the interleavers, achieve a target post-FEC bit error rate (BER). This combination of interleavers and BCH codes has very low implementation complexity. In addition, our approach is straightforward, requiring only short pre-FEC simulations to parameterize a model, based on which we select codes analytically. We aim to correct a pre-FEC BER of around $$10^{-3}$$10-3. We evaluate the accuracy of our approach using numerical simulations. For a target post-FEC BER of $$10^{-6}$$10-6, codes selected using our method result in BERs around 3$$\times $$× target and achieve the target with around 0.2 dB extra signal-to-noise ratio.

Improved BER Performance of Real-Time DDO-OFDM Systems Using Interleaved Reed–Solomon Codes

In this letter, we have experimentally demonstrated a Reed–Solomon (RS) coded and multiple symbol interleaved real-time orthogonal frequency-division multiplexing (OFDM) signals transmission over a single mode fiber (SMF)-based direct-detection optical system. To the best of our knowledge, this is the first time to realize the forward error correction (FEC) in real-time optical OFDM systems. The bit error rate (BER) performances of RS-coded OFDM signals are investigated in a low-cost directly modulated laser-based direct-detection optical transmission system. The experimental results show that after 25.26-km SMF-28 transmission of the real-time RS-coded and symbol interleaved optical OFDM signals, a post-FEC BER less than $1\times 10^{-8}$ is successfully achieved by using RS code with 7% overhead for a pre-FEC BER of $1\times 10^{-3}$ .

Demonstration of Software-Reconfigurable Real-Time FEC-Enabled 4/16/64-QAM-OFDM Signal Transmission in an X-Band RoF System

In this paper, to the best of our knowledge, this is the first time that the generation and reception of online software-reconfigurable real-time 4/16/64-quadrature amplitude modulation (QAM)-encoded and forward error correction (FEC)-enabled orthogonal frequency-division multiplexing (OFDM) signals over X-band fiber-over-radio-based upstream links has been demonstrated. A Reed–Solomon code, i.e., RS (288, 256), with symbol interleaving/deinterleaving is applied to enhance bit-error-rate (BER) performance. The experimental results show that the 4/16/64-QAM-OFDM signals with net data rates of 1.6/3.2/4.8-Gb/s after 1.5-m wireless and 2.26-km SMF-28 transmission with BERs below the hard-decision FEC threshold of 3.8e-3 can be successfully achieved without using FEC. With the help of symbol-interleaved RS code, a post-FEC BER below 1e-8 can be observed when the pre-FEC BER is around 1e-3.

Replacing the Soft-Decision FEC Limit Paradigm in the Design of Optical Communication Systems

The FEC limit paradigm is the prevalent practice for designing optical communication systems to attain a certain bit error rate (BER) without forward error correction (FEC). This practice assumes that there is an FEC code that will reduce the BER after decoding to the desired level. In this paper, we challenge this practice and show that the concept of a channel-independent FEC limit is invalid for soft-decision bit-wise decoding. It is shown that for low code rates and high-order modulation formats, the use of the soft-decision FEC limit paradigm can underestimate the spectral efficiencies by up to 20%. A better predictor for the BER after decoding is the generalized mutual information, which is shown to give consistent post-FEC BER predictions across different channel conditions and modulation formats. Extensive optical full-field simulations and experiments are carried out in both the linear and nonlinear transmission regimes to confirm the theoretical analysis.

Post-FEC Performance of Nondifferential Transmission in the Presence of Phase Noise

We assess the impact of phase noise on the bit error rate (BER) after forward error correction (FEC) decoding (post-FEC BER) and on the error distribution in the case of differential and nondifferential (ND) transmission. We show that the use of pilot symbols enables reliable ND transmission and relaxes the requirements on bit-interleaving, and therefore, on the end-to-end latency. Further, we provide guidelines for the design and choice of carrier recovery and FEC parameters to obtain good performance in typical scenarios. All presented results are achieved through the use of a high-speed system emulation platform based on field-programmable gate arrays and thus achieve high reliability even for very low post-FEC BER.

Four-Dimensional Coded Modulation with Bit-Wise Decoders for Future Optical Communications

Coded modulation (CM) is the combination of forward error correction (FEC) and multilevel constellations. Coherent optical communication systems result in a four-dimensional (4D) signal space, which naturally leads to 4D-CM transceivers. A practically attractive design paradigm is to use a bit-wise decoder, where the detection process is (suboptimally) separated into two steps: soft-decision demapping followed by binary decoding. In this paper, bit-wise decoders are studied from an information-theoretic viewpoint. 4D constellations with up to 4096 constellation points are considered. Metrics to predict the post-FEC bit-error rate (BER) of bit-wise decoders are analyzed. The mutual information is shown to fail at predicting the post-FEC BER of bit-wise decoders and the so-called generalized mutual information is shown to be a much more robust metric. For the suboptimal scheme under consideration, it is also shown that constellations that transmit and receive information in each polarization and quadrature independently (e.g., PM-QPSK, PM-16QAM, and PM-64QAM) outperform the best 4D constellations designed for uncoded transmission. Theoretical gains are as high as 4 dB, which are then validated via numerical simulations of low-density parity check codes.

Interleavers and BCH Codes for Coherent DQPSK Systems With Laser Phase Noise

The relatively high phase noise of coherent optical systems poses unique challenges for forward error correction (FEC). In this letter, we propose a novel semianalytical method for selecting combinations of interleaver lengths and binary Bose-Chaudhuri-Hocquenghem (BCH) codes that meet a target post-FEC bit error rate (BER). Our method requires only short pre-FEC simulations, based on which we design interleavers and codes analytically. It is applicable to pre-FEC BER $\sim 10^{-3}$ , and any post-FEC BER. In addition, we show that there is a tradeoff between code overhead and interleaver delay. Finally, for a target of $10^{-5}$ , numerical simulations show that interleaver-code combinations selected using our method have post-FEC BER around $2\times $ target. The target BER is achieved with 0.1 dB extra signal-to-noise ratio.

Dimensioning BCH Codes for Coherent DQPSK Systems With Laser Phase Noise and Cycle Slips

Forward error correction (FEC) plays a vital role in coherent optical systems employing multi-level modulation. However, much of coding theory assumes that additive white Gaussian noise (AWGN) is dominant, whereas coherent optical systems have significant phase noise (PN) in addition to AWGN. This changes the error statistics and impacts FEC performance. In this paper, we propose a novel semianalytical method for dimensioning binary Bose-Chaudhuri-Hocquenghem (BCH) codes for systems with PN. Our method involves extracting statistics from pre-FEC bit error rate (BER) simulations. We use these statistics to parameterize a bivariate binomial model that describes the distribution of bit errors. In this way, we relate pre-FEC statistics to post-FEC BER and BCH codes. Our method is applicable to pre-FEC BER around 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-3</sup> and any post-FEC BER. Using numerical simulations, we evaluate the accuracy of our approach for a target post-FEC BER of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-5</sup> . Codes dimensioned with our bivariate binomial model meet the target within 0.2-dB signal-to-noise ratio.

Impact of Nonideal Phase Reference on Soft Decoding of Differentially Encoded Modulation

The performance of coded modulation schemes based on the concatenation of a forward-error correction (FEC) code and differentially encoded quadrature amplitude modulations (QAMs) in the presence of Wiener phase noise is considered. Log-likelihood ratios, required to perform soft decoding, are first derived for the case in which the phase of the received carrier is assumed to be perfectly known, and then used to obtain the performance for nonideal reference of phase. Simulation results of the post-FEC bit-error rate are presented for a near-Shannon limit low-density parity check code with an overhead of 20%. The performance loss, compared to ideal coherent demodulation, is evaluated for two different linewidths of the phase noise in case of quadrature-phase shift keying and 16-QAM transmission.