Symbol Probabilities Research Articles

A cascaded bit-weighted distribution matching for LDPC-coded PS-64QAM DMT in a high-speed Unamplified IM-DD system

We experimentally demonstrate the generation and transmission of probabilistic shaping (PS) 64-ary quadrature amplitude modulation (64-QAM) discrete multi-tone (DMT) signal with low-density parity-check (LDPC)-coded modulation based on two-stage bit-weighted distribution matching (TS-BWDM) in a high-speed unamplified intensity modulation and direct detection system. The TS-BWDM scheme is based on a two-stage bit-weighted inversion algorithm, which has the advantages of no complex multiplication and division operations and low hardware implementation complexity. As the key content of TS-BWDM, before FEC coding, the bit-weighted intervention operation is used to change the bit probability of a specific sequence in the parallel binary sequence. The two-stage bit-weighted inversion operation and one-stage weight-labeling bit insertion is performed for achieving the target symbol probability distribution. As a result, it does not add a significant amount of extra redundant bits. Additionally, the high-accuracy training symbols-based sampling frequency offset (SFO) estimation method and digital interpolation are used to compensate SFO. In our experiments, the TS-BWDM-based PS-64QAM DMT signals with a net rate of 157.3-Gb/s transmission over 10-km non-zero dispersion- shifted fiber can be achieved. The system performance is investigated under two LDPC code rates (3/4 and 9/10) and three PS parameter values (k = 3, 5 and 9). The experimental results show that the receiver power sensitivity and the system fiber nonlinear effect tolerance can be significantly improved compared with uniformly-distributed signals.

Performance Analysis of Multiple-Antenna Ambient Backscatter Systems at Finite Blocklengths

This paper analyzes the maximal achievable rate for a given blocklength and maximal error probability over a multiple-antenna ambient backscatter channel. The result consists of a finite blocklength channel coding achievability bound and a converse bound for the legacy system with finite alphabet constraints and multiple-input-multiple-output based on the Neyman-Pearson test, the Berry-Esseen theorem, and the Mellin transform. Then, we derive the closed-form expression of the mutual information and the information variance to reduce the complexity of the computation. By applying the low-complexity ML detection, the relation between the maximal error probability of the RF source signal and the average error probability of the tag symbol with respect to the blocklength is proposed. Finally, numerical evaluation of these bounds shows fast convergence to the maximal achievable rate as the blocklength increases and also proves that the information variance is an accurate measure of the backoff from the maximal achievable rate due to finite blocklength.

Probabilistic constellation shaping-aided underwater acoustic communication with vector approximate message passing iterative equalization.

A probabilistic constellation shaping (PCS)-aided single-carrier transceiver is proposed to improve spectral efficiency for underwater acoustic (UWA) communications. At the transmitter, the information bits are input into a distribution matcher followed by a systematic binary encoder, which yields a sequence of sign bits with a uniform distribution and a sequence of amplitude bits with a non-uniform distribution. Based on these two sequences, the PCS is then realized by mapping coded bits onto a quadrature amplitude modulation constellation. At the receiver, an improved frequency-domain turbo equalizer based on the vector approximate message passing (VAMP) is proposed for the PCS-UWA communication system to eliminate the multipath interference. It exploits the a priori symbol probability information benefiting from the PCS at the beginning of the turbo iteration and over the self-iteration of the VAMP soft equalizer, improving the symbol detection performance. Finally, the first experimental demonstration of a deep-sea PCS-UWA communication system and numerical simulation of shallower water are presented. The experimental results reveal the PCS-UWA communication system significantly outperforms traditional systems with no PCS. Also, the proposed receiver is superior to the classical adaptive turbo equalizer based on the improved proportionate normalized least mean square algorithm even with data reuse.

Transformer-Based Detection for Highly Mobile Coded OFDM Systems.

This paper is concerned with mobile coded orthogonal frequency division multiplexing (OFDM) systems. In the high-speed railway wireless communication system, an equalizer or detector should be used to mitigate the intercarrier interference (ICI) and deliver the soft message to the decoder with the soft demapper. In this paper, a Transformer-based detector/demapper is proposed to improve the error performance of the mobile coded OFDM system. The soft modulated symbol probabilities are computed by the Transformer network, and are then used to calculate the mutual information to allocate the code rate. Then, the network computes the codeword soft bit probabilities, which are delivered to the classical belief propagation (BP) decoder. For comparison, a deep neural network (DNN)-based system is also presented. Numerical results show that the Transformer-based coded OFDM system outperforms both the DNN-based and the conventional system.

Real-time demonstration of a low-complexity PS scheme for 16QAM-DMT signals in an IM-DD system.

We experimentally demonstrated a low-complexity probabilistic shaping (PS) 16-ary quadrature amplitude modulation (16QAM) scheme based on intra-symbol bit-weighted distribution matching (Intra-SBWDM) for discrete multi-tone (DMT) symbols with field-programmable gate array implementation in an intensity modulation and direct detection (IM-DD) system. Different from the traditional PS scheme such as Gallager many-to-one mapping, hierarchical distribution matching and constant composition distribution matching, the Intra-SBWDM scheme with lower computation and hardware complexity does not need to continuously refine the interval to find the target symbol probability, nor does it need a look-up table, so it does not introduce a large number of extra redundant bits. In our experiment, four PS parameter values (k = 4, 5, 6, and 7) are investigated in a real-time short-reach IM-DD system. 31.87-Gbit/s net bit PS-16QAM-DMT (k = 4) signal transmission is achieved. The results show that the receiver sensitivity in terms of the received optical power of the real-time PS scheme based on Intra-SBWDM (k = 4) over OBTB/20 km standard single-mode fiber can be improved by about 1.8/2.2 dB at the bit error rate (BER) of 3.8 × 10-3, compared to the uniformly-distributed DMT. In addition, the BER is steadily lower than 3.8 × 10-3 during a one-hour measurement for PS-DMT transmission system.

VLSI Design of a High-Performance Multicontext MQ Arithmetic Coder

The MQ arithmetic coding, which is an adaptive arithmetic coding developing from Q coding, has become a major throughput bottleneck of JPEG2000 compression due to its inherent serial operations. To overcome the bottleneck, this brief proposes a high-performance hardware architecture of the multicontext MQ coder. The proposed architecture is capable of concurrent coding for two adjacent more probable symbols (MPSs). Performance analysis results show that the proposed coder consumes 1.61 CXD pairs per cycle and achieves a throughput of 506.93 MSymbols/s under 0.5 bpp bit rate. The proposed architecture not only achieves high throughput, but also maintains both low hardware utilization and low power consumption. Compared with the state-of-the-art two-context coder, the figure of merit (FoM) is increased by 38%. Compared with the single-context coder, the power-delay product (PDP) is reduced by 49%.

Introduction to Symbolic 2-Plithogenic Probability Theory

In this paper we present for the first time the concept of symbolic plithogenic random variables and study its properties including expected value and variance. We build the plithogenic formal form of two important distributions that are exponential and uniform distributions. We find its probability density function and cumulative distribution function in its plithogenic form. We also derived its expected values and variance and the formulas of its random numbers generating. We finally present the fundamental form of plithogenic probability density and cumulative distribution functions. All the theorems were proved depending on algebraic approach using isomorphisms. This paper can be considered the base of symbolic plithogenic probability theory.

Analysis of the Limitations of Further Improvement of the Efficiency of VVC-CABAC

Hybrid video compression plays an invaluable role in digital video transmission and storage services and systems. It performs several-hundred-fold reduction in the amount of video data, which makes these systems much more efficient. An important element of hybrid video compression is entropy coding of the data. The state-of-the-art in this field is the newest variant of the Context-based Adaptive Binary Arithmetic Coding (CABAC) entropy compression algorithm which recently became part of the new Versatile Video Coding (VVC) technology. This work is a part of research that is currently underway to further improve the CABAC technique. This paper provides analysis of the potential for further improvement of the CABAC by more accurate calculation of probabilities of data symbols. The CABAC technique calculates those probabilities by the use of idea of the two-parameters hypothesis. For the needs of analysis presented in this paper, an extension of the aforementioned idea was proposed which consists of three- and four-parameters hypothesis. In addition, the paper shows the importance of proper calibration of parameter values of the method on efficiency of data compression. Results of experiments show that for the considered in the paper variants of the algorithm improvement the possible efficiency gain is at levels 0.11% and 0.167%, for the three- and four-parameter hypothesis, respectively.

Memory-aware end-to-end learning of channel distortions in optical coherent communications.

We implement a new variant of the end-to-end learning approach for the performance improvement of an optical coherent-detection communication system. The proposed solution enables learning the joint probabilistic and geometric shaping of symbol sequences by using auxiliary channel model based on the perturbation theory and the refined symbol probabilities training procedure. Due to its structure, the auxiliary channel model based on the first order perturbation theory expansions allows us performing an efficient parallelizable model application, while, simultaneously, producing a remarkably accurate channel approximation. The learnt multi-symbol joint probabilistic and geometric shaping demonstrates a considerable bit-wise mutual information gain of 0.47 bits/2D-symbol over the conventional Maxwell-Boltzmann shaping for a single-channel 64 GBd transmission through the 170 km single-mode fiber link.

Statistical Arithmetic Coding Algorithm Adaptive to Correlation Properties of Wavelet Transform Coefficients

It is shown that, in order to increase the compression ratio in the course of statistical arithmetic coding, it is necessary to take into account the conditional probabilities of code symbols when the preceding symbols appear. The problem of obtaining the location of the most significant symbols when encoding the current symbol is solved by calculating the autocorrelation function of the encoded symbols. An algorithm for arithmetic coding and decoding is provided, which takes into account the dependencies between the coefficients of the wavelet transform and the results of modeling its operation.

Probabilistic shaping PAM8 based on Huffman coding distribution matching in optical interconnection

We propose and experimentally verify a Huffman coding distribution matching (HCDM) scheme in intensity modulation and direct-detection (IM/DD) optical interconnection applying low-density parity-check-coded (LDPC) probabilistic shaping 8-level pulse amplitude modulation (PS-PAM8). Not only superior to that of conventional PS methods in view of arithmetic distribution matching (ADM) as well as constant composition distribution matching (CCDM) with much lower computational complexity, the HCDM can also enable arbitrary symbol probability distribution (SPD) by utilizing an optimal Huffman binary tree. The binomial coefficients required for SPD can be precomputed and stored in a lookup table. The experimental results show that the HCMD PS-PAM8 shows remarkably promoted receiver power sensitivity and improved system tolerance to optical fibre nonlinear effects compared to the uniformly distributed signal with the same effective net rate.

EMF-Aware Probabilistic Shaping Design for Hardware-Distorted Communication Systems

The fifth-generation cellular network requires dense installation of radio base stations (BS) to support the ever-increasing demands of high throughput and coverage. The ongoing deployment has triggered some health concerns among the community. To address this uncertainty, we propose an EMF-aware probabilistic shaping design for hardware-distorted communication systems. The proposed scheme aims to minimize human exposure to radio frequency (RF) radiations while achieving the target throughput using probabilistic shaping. The joint optimization of the transmit power and nonuniform symbol probabilities is a non–convex optimization problem. Therefore, we employ alternate optimization and successive convex approximation to solve the subsequent problems. Our findings reveal a significant reduction in the users’ exposure to EMF while achieving the requisite quality of service with the help of probabilistic shaping in a hardware-distorted communication system.

Unity-Rate Coding Improves the Iterative Detection Convergence of Autoencoder-Aided Communication Systems

A forward error correction (FEC) and unity-rate coded (URC) autoencoder (AE)-assisted communication system is proposed for the first time, which relies on soft iterative decoding for attaining a vanishingly low error probability. The AE-demapper is specifically designed for directly calculating the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">extrinsic</i> logarithmic likelihood ratios (LLRs), which can be directly entered into the URC decoder for soft iterative decoding. This avoids the potential degradation due to the conversion of symbol probabilities to bit LLRs. A comprehensive capacity analysis of the AE is performed, which demonstrates the capacity advantage of the AE-aided constellation design over its conventional quadrature amplitude modulation (QAM)/ phase shift keying (PSK) counterpart. Furthermore, we carry out its EXtrinsic Information Transfer (EXIT) chart analysis, which indicates that as a benefit of our URC, the EXIT curve always reaches the [1,1] point of perfect convergence, leading to a vanishingly low error probability. More explicitly, our bit error ratio (BER) and block error ratio (BLER) results demonstrate that the proposed FEC-URC-AE system achieves significant iterative gains both in additive white Gaussian noise (AWGN) and Rayleigh channels, outperforming both its model-based FEC-AE and its conventional coded QAM/QPSK counterparts.

Generalized Good-Turing Improves Missing Mass Estimation

Consider a finite sample from an unknown distribution over a countable alphabet. The missing mass refers to the probability of symbols that do not appear in the sample. Estimating the missing mass is a basic problem in statistics and related fields, which dates back to the early work of Laplace, and the more recent seminal contribution of Good and Turing. In this article, we introduce a generalized Good-Turing (GT) framework for missing mass estimation. We derive an upper-bound for the risk (in terms of mean squared error) and minimize it over the parameters of our framework. Our analysis distinguishes between two setups, depending on the (unknown) alphabet size. When the alphabet size is bounded from above, our risk-bound demonstrates a significant improvement compared to currently known results (which are typically oblivious to the alphabet size). Based on this bound, we introduce a numerically obtained estimator that improves upon GT. When the alphabet size holds no restrictions, we apply our suggested risk-bound and introduce a closed-form estimator that again improves GT performance guarantees. Our suggested framework is easy to apply and does not require additional modeling assumptions. This makes it a favorable choice for practical applications. Supplementary materials for this article are available online.

Early detection of lean blowout in a combustor using symbolic analysis of colour images

Recently, lean combustion has been accepted as one of the preferable modes to run both small and large scale energy sectors as it reduces the level of NOx (oxides of nitrogen) emissions. However, ultra-lean combustion is more susceptible to lean blowout (LBO) which significantly reduces engine life and operating reliability. Therefore, early detection of LBO is important so that operator can get sufficient lead time to initiate precautionary actions. The current investigation proposes a novel technique based on the conversion of flame colour matrix into symbolic strings. The generation of symbolic strings is done based on the partitioning of the hue space obtained from transformation of RGB (Red-Green-Blue) to HSV (Hue-Saturation-Value) using uniform partitioning method. The partitions are determined at the reference state and kept fixed for other equivalence ratios (anomalous or non-reference states). Thus, the hue value at each pixel of the image is assigned a symbol depending on its belonging to a particular partition. Thereafter, an array of symbolic state probability vector at reference combustion state is determined based on the probability of pixel hue value being in a particular symbolic state or symbol (number of symbolic states is considered as 8). Also, for other anomalous or non-reference combustion states, symbolic state probability vectors are estimated using the partition formed at the reference state. The difference in symbolic state probability vectors between reference and anomalous combustion states gives the anomaly measure, which is used here to detect the flame blowout. The early increasing trend of anomaly towards the blowout for both premixed and partially premixed flames indicate that the measure can be helpful to detect the approach towards LBO. The estimation of anomaly is shown for using the flame images of different quantities. The random selection of five images from a flame video shows to be very effective in capturing the trend of anomaly which also ensures low computational effort. Thus, the tool based on symbolic colour image analysis (SCIA) may also be helpful in the mitigation of LBO.

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.

Measurement of the Probability of a Binary Symbol «0» Erasing in a Single-Photon Asynchronous Communication Channel with a Receiver Based on a Photon Counter

Receiving modules of single-photon communication channels should provide the least loss of transmitted information when measuring low-power optical signals. In this regard, it is advisable to use photon counters. They are highly sensitive, but are characterized by data logging errors. Therefore, the purpose of this work was to investigate the effect of the intensity of the recorded optical radiation during the transmission of binary symbols «0» on the probability of erasing these symbols in a single-photon communication channel containing a photon counter based on an avalanche photodetector as a receiving module with a passive avalanche suppression scheme.The lower and upper threshold levels of pulses recorded at the output of the photon counter, as well as the statistical distributions of the mixture of the number of dark and signal pulses at the output of the photon counter when registering binary symbols «0» Pst0( N ) and «1» Pst1( N ) were determined. For this, a technique was used to reduce information loss. As a result, the minimum probability of erasing binary symbols «0» P(–/0) was achieved.The performed experimental results showed that to achieve the minimum probability of erasing binary symbols «0» P(–/0) = 0,11·10−2, it is important to select not only the intensity of the used optical radiation J , but also the supply voltage of the avalanche photodetector U, at which the dead time of the photon counter is −2 minimal, and its quantum detection efficiency is maximum: J0 ≥ 98,94·10−2 rel. units and U = 52,54 V.

Threshold Detection Scheme Based on Parametric Distribution Fitting for Optical Fiber Channels

Background: When data is transmitted by encoding it in the amplitude of the transmitted signal, such as in Amplitude Shift Keying, the receiver makes a decision on the transmitted data by observing the amplitude of the received signal. Objective: Depending on the level of the received signal, the receiver has to decode the received data, with minimum probability of error. For binary transmission, this is achieved by choosing a decision threshold. Methods: A threshold detection mechanism is evaluated in this paper, which uses parametric probability distribution fitting to the received signal. Results: Based on our earlier work on visible light communication, the idea is extended to a fiber optic channel and it is found that in a fiber optic channel, the threshold value obtained by fitting a Rayleigh distribution to the received data results in error probability approaching zero. Conclusion: It is found that the proposed threshold estimation method works well for both equal as well as unequal apriori probabilities of the binary symbols. The proposed method of threshold detection is adaptive when the apriori probabilities for 1’s and 0’s change.

Low-Complexity EP Receiver Based on Location-Aware and Reliability-Aware Strategy

The expectation propagation (EP) algorithm has received significant attention recently due to its good performance and manageable computational complexity for large-scale inference problems, such as detection for massive multiple-input multiple-output (MIMO) systems. Its computational complexity depends mainly on how messages are represented for computation and passing. In this letter, we propose a low-complexity EP algorithm based on location-aware and reliability-aware strategy, shortened as LR-EP, to efficiently implement the message computation and passing. At each iteration, only the most probable symbol candidates are selected to perform message computation based on a low-complexity quadrant detection and predetermined lookup table. In the iterative process, reliable variable nodes with sufficiently high posterior probabilities are kept intact, thus reducing the less efficient message passing. Numerical results illustrate that the proposed LR-EP receiver reduces computational complexity significantly compared to the conventional EP receiver with negligible performance degradation.

Symbolic analysis of bursting dynamical regimes of Rulkov neural networks

Neurons modeled by the Rulkov map display a variety of dynamic regimes that include tonic spikes and chaotic bursting. Here we study an ensemble of bursting neurons coupled with the Watts-Strogatz small-world topology. We characterize the sequences of bursts using the symbolic method of time-series analysis known as ordinal analysis, which detects nonlinear temporal correlations. We show that the probabilities of the different symbols distinguish different dynamical regimes, which depend on the coupling strength and the network topology. These regimes have different spatio-temporal properties that can be visualized with raster plots.