Fixed-rate Coding Research Articles

An Asymptotically Optimal Two-Part Fixed-Rate Coding Scheme for Networked Control With Unbounded Noise

It is known that under fixed-rate information constraints, adaptive quantizers can be used to stabilize an open-loop-unstable linear system on Rn driven by unbounded noise. These adaptive schemes can be designed so that they have near-optimal rate, and the resulting system will be stable in the sense of having an invariant probability measure, or ergodicity, as well as boundedness of the state second moment. Although structural results and information theoretic bounds of encoders have been studied, the performance of such adaptive fixed-rate quantizers beyond stabilization has not been addressed. In this paper, we propose a two-part adaptive (fixed-rate) coding scheme that achieves state second moment convergence to the classical optimum (i.e., for the fully observed setting) under mild moment conditions on the noise process. The first part, as in prior work, leads to ergodicity (via positive Harris recurrence) and the second part ensures that the state second moment converges to the classical optimum at high rates. These results are established using an intricate analysis which uses random-time state-dependent Lyapunov stochastic drift criteria as a core tool.

Experimental Demonstration of Rate-Adaptive Concatenated Codes for Flexible Optical Networks

We experimentally demonstrate a rate-adaptive concatenated forward error correction (FEC) scheme to minimize the information rate loss due to varying channel conditions. The concatenated coding scheme is based on inner soft-decision polar code and outer hard-decision staircase code. An experiment in a wavelength-division multiplexed (WDM) system with different SNR per WDM channel is performed by employing concatenated codes with various coded modulation schemes over a target distance of 140km which is relevant for extending the reach of data center interconnects. Experimental results show that the achieved data rate of the WDM system can be improved up to 37.6% with the proposed coding scheme w.r.t. the fixed code rate ZR FEC.

Lattice-Based Minimum-Distortion Data Hiding

Lattices have been conceived as a powerful tool for data hiding. While conventional studies and applications focus on achieving the optimal robustness versus distortion tradeoff, in some applications such as data hiding in medical/physiological signals, the primary concern is to achieve a minimum amount of distortion to the cover signal. In this paper, we revisit the celebrated quantization index modulation (QIM) scheme and propose a minimum-distortion version of it, referred to as MD-QIM. The crux of MD-QIM is to move the data point to only the boundary of the Voronoi region of the lattice point indexed by a message, which suffices for subsequent correct decoding. At any fixed code rate, the scheme achieves the minimum amount of distortion by sacrificing the robustness to the additive white Gaussian noise (AWGN) attacks. Simulation results confirm that our scheme significantly outperforms QIM in terms of mean square error (MSE), peak signal to noise ratio (PSNR) and percentage residual difference (PRD).

Stabilizing a System With an Unbounded Random Gain Using Only Finitely Many Bits

We study the stabilization of a linear control system with an unbounded random system gain where the controller must act based on a rate-limited observation of the state. More precisely, we consider the system X <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n+1</sub> =A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> X <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> +W <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> -U <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> , where the A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> 's are drawn independently at random at each time n from a known distribution with unbounded support, and where the controller receives at most R bits about the system state at each time from an encoder. We provide a time-varying achievable strategy to stabilize the system in a second-moment sense with fixed, finite R. While our previous result provided a strategy to stabilize this system using a variable-rate code, this work provides an achievable strategy using a fixed-rate code. The strategy we employ to achieve this is time-varying and takes different actions depending on the value of the state. It proceeds in two modes: a normal mode (or zoom-in), where the realization of A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> is typical, and an emergency mode (or zoom-out), where the realization of A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> is exceptionally large. To analyze the performance of the scheme we construct an auxiliary sequence that bounds the state X <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> , and then bound auxiliary sequence in both the zoom-in and zoom-out modes.

Design and Analysis of LT Codes With a Reverse Coding Framework

In this paper, an improved LT code with a reverse coding framework is designed to reduce the error floor caused by low-degree information nodes. For the proposed coding scheme, a well-designed threshold is used to mark the information nodes whose degrees are less than the threshold, and these nodes will be coded reversely to connect to enough candidate check nodes. To design the optimal threshold, firstly, the information degree distribution and the check degree distribution of the improved LT code are deduced. Then, the parameter extrinsic information gain-loss-ratio (GLR) is designed to evaluate the convergence behavior of the improved LT code. Finally, the ‘slow increase region’ of the GLR is set, and the boundary value of this region is used to deduce the optimal threshold which matches with the channel state information (CSI) and decoding overhead. To make the proposed LT code not limited to a fixed code rate, we further modify the proposed scheme. The segment coding method is used to generate a redundant generator matrix, and the check nodes corresponding to this matrix can be transmitted independently and are not limited to a fixed number. Furthermore, the connection relationship between information nodes and check nodes can be easily recorded, which improves the decoding efficiency. The advantages of the improved LT code are that the degree distributions can be formulated, the convergence behavior can be predicted, and the lowest information degree can be adjusted. Simulation results show that the improved LT code can reduce the error floor by up to 4 orders of magnitude. Besides, the designed LT code outperforms the existing LT codes in literature in terms of bit error rate (BER) performance.

Multiple-rate error-correcting coding scheme

Error-correcting codes that can effectively encode and decode messages of distinct lengths while maintaining a constant blocklength are considered. It is known conventionally that a k-dimensional block code of length n defined over $$\texttt {GF}(q^{n})$$ is designed to encode a k-symbol user data in to an n-length codeword, resulting in a fixed-rate coding. In contrast, considering $$q=p^{\lambda }$$, this paper proposes two coding procedures (for the cases of $$\lambda =k$$ and $$\lambda =n$$) each deriving a multiple-rate code from existing channel codes defined over a composite field $$\texttt {GF}(q^{n})$$. Formally, the proposed coding schemes employ $$\lambda$$ codes $${\mathcal {C}}_{1}(\lambda , 1), {\mathcal {C}}_{2}(\lambda , 2), \ldots , {\mathcal {C}}_{\lambda }(\lambda , \lambda )$$ defined over $$\texttt {GF}(q)$$ to encode user messages of distinct lengths and incorporate variable-rate feature. Unlike traditional block codes, the derived multiple-rate codes of fixed blocklength n can be used to encode and decode user messages $$\mathbf{m}$$ of distinct lengths $$|\mathbf{m}| = 1, 2, \ldots , k, k+1, \ldots , kn$$, thereby supporting a range of information rates—inclusive of the code rates $$1/n^{2}, 2/n^{2},\ldots , k/n^{2}$$ and $$1/n, 2/n, \ldots , k/n$$ ! A simple decoding procedure to the derived multiple-rate code is also given; in that, orthogonal projectors are employed for the identification of encoded user messages of variable length.

Rateless-Code-Based Secure Cooperative Transmission Scheme for Industrial IoT

Integrating the Internet-of-Things (IoT) paradigm with industrial wireless sensor networks (IWSNs) has led to a technological evolution known as Industrial IoT (IIoT). To ensure reliable, secure, and low-latency communication in IIoT applications like healthcare, we use rateless codes (RCs) for forward error correction (FEC) at the physical layer (PHY) in a cooperative delay-constrainted environment. Probability of violation (Pr <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">vio</sub> ) is used as a performance metric which consists of: 1) reliability outage probability, i.e., the probability with which the information cannot be successfully decoded within the time constraint T and 2) information intercept probability, which reflects the secrecy performance. Since RC can adjust the rate on the fly, we analyze the performance when relay transmits a fraction of the message and the receiver does energy accumulation (EA). We derive a closed-form expression of Prvio in a single-relay system at a high signal-to-noise ratio (SNR) when the receiver either accumulates energy or mutual information (MI) from the received signals. Furthermore, Pr <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">vio</sub> of various relay selection schemes is evaluated in a multirelay system. Such kind of analysis in which relay selection is done using RC for PHY-FEC is first of its kind and it is compared to fixed-rate codes. Overall, results show the significant benefits of PHY-RC for guaranteeing the Quality-of-Service (QoS) requirements essential in IIoT applications.

On the Age of Information in Wireless Networks Using Rateless Codes

The information freshness, quantified by the emerging idea of Age of Information (AoI), has recently received widespread attentions from both academia and industry. However, most prior works on AoI considered a single source-destination pair and provided the performance analysis merely through queueing-theoretic models, which do not lend themselves to the analysis of large-scale wireless networks. This paper focuses on the AoI analysis in a large-scale setting using random spatial models in stochastic geometry. Specifically, we first adopt rateless coding for packet delivery to improve the transmission reliability while reducing the transmission delay. Then, to quantify the enhancement of the information freshness by rateless coding, we derive the average peak age of information (PAoI) in Poisson bipolar and cellular networks, which characterizes the maximum value of AoI immediately before an update packet is received accounting for the spatial variation of network nodes and temporal dynamics of service packets. With these results, we further give the conditions that make the average PAoI to be finite in each type of the network. Numerical results show the significant benefits of rateless codes relative to the commonly used fixed-rate codes in terms of the PAoI and the feasible region of system parameters such as the traffic arrival rate, transmit probability, etc., in Poisson bipolar and cellular networks.

The End-to-End Performance of Rateless Codes in Poisson Bipolar and Cellular Networks

Rateless coding is a promising forward error correction technique to meet both delay and reliability requirements of emerging wireless applications. However, existing works on rateless codes mostly considered finite networks or ignored the traffic dynamics. In this paper, we present a comprehensive investigation of the end-to-end performance for rateless codes in Poisson bipolar and cellular networks. Specifically, we propose the notion of the end-to-end success probability , which is the success probability given an end-to-end delay requirement, to jointly evaluate the delay performance and transmission reliability of rateless codes. To fully characterize the end-to-end delay, we divide it into two parts, namely the packet waiting time and the transmission time, and provide tractable yet accurate approximations to their statistical distributions. Compared with the previous works, the proposed general framework and the end-to-end performance metric help obtain insights on the role of scheduling, queueing, and coding scheme in practical radio access networks. The approximations are verified to be effective and reliable through simulations. Overall, the results show the significant benefits of rateless codes relative to the fixed-rate codes in terms of the transmission reliability with an end-to-end delay requirement.

Orthogonal coding for noise immunity’s increase with the fixed code rate

The objective of the research, which results are presented in this paper, consists in a development and a research of the offered by the author orthogonal coding as a way of noise immunity’s increase using the example of the information communication systems with phase-shift keying. It is shown that the orthogonal coding allows one to provide the required quality of communication with a smaller energy cost. The energy gain in a signal-to-noise ratio is provided by more effective use of energy of transmitted signals with the fixed code rate without increase in complexity and cost of transmitting/receiving devices. The orthogonal coding is an analogue of convolutional coding over the rational numbers’ field. Properties of orthogonality of proposed codes allow one to strengthen significance of a useful signal with simultaneous weakening of influence of AWGN. Such increase of a signal-to-noise ratio is preserved in various fading channels. Technical realization of orthogonal coding is characterized by low complexity: at each step a decoding process is reduced to calculation of several dot products and comparison with fixed (zero, in this case) threshold. For this reason, the offered way of coding and design of receiving and transmitting devices can be implemented in various communication systems.

Overlapped LT codes over the binary erasure channel: analysis and design

Fountain codes are promising forward error correction codes suitable for broadcast and multicast applications where many users with different capabilities are involved in the system. Fixed-rate codes need a priori information about each channel to decide the best code rate and the code rate is usually chosen according to the worst channel to avoid an outage. Fountain codes are agnostic to the channel conditions and rateless. The rate is realised based on the channel conditions for users separately. Overlapped fountain codes were previously invented over the binary erasure channel to provide more degrees of freedom and better trade-offs for the rateless-coded system parameters. In this study, the authors show via analysis and simulations that overlapped fountain codes need fewer decoding steps in comparison with the conventional fountain codes. For example, overlapped fountain codes reduce the number of decoding steps/iterations of belief propagation decoder by 29 % even at a source length k = 256 . They optimise the overlap selection probability of the overlapped fountain codes to maximise the code rate and/or minimise the complexity. At source length, k = 256 , the proposed analytical and simulation studies show that the highest code rate and lowest average complexity are achieved at an overlap selection probability p = 0.6 .

False-Accept/False-Reject Trade-Offs for Ensembles of Biometric Authentication Systems

Biometric authentication systems, based on secret key generation, work as follows. In the enrollment stage, an individual provides a biometric signal that is mapped into a secret key and a helper message, the former being prepared to become available to the system at a later time (for authentication), and the latter is stored in a public database. When an authorized user requests authentication, claiming his/her identity to be one of those of the subscribers, he/she has to provide a biometric signal again, and then the system, which retrieves also the helper message of the claimed subscriber, produces an estimate of the secret key that is finally compared to the secret key of the claimed user. In the case of a match, the authentication request is approved; otherwise, it is rejected. Evidently, there is an inherent tension between two desired, but conflicting, properties of the helper message encoder; on the one hand, the encoding should be informative enough concerning the identity of the real subscriber, in order to approve him/her in the authentication stage, but on the other hand, it should not be too informative, as otherwise, unauthorized imposters could easily fool the system and gain access. A good encoder should then trade off the two kinds of errors: the false reject (FR) error and the false accept (FA) error. In this paper, we investigate trade-offs between the random coding FR error exponent and the best achievable FA error exponent. We compare two types of ensembles of codes: fixed-rate codes and variable-rate codes, and we show that the latter class of codes offers considerable improvement compared to the former. In doing this, we characterize ensemble-optimal rate functions for both types of codes. We also examine the effect of privacy leakage constraints for both fixed-rate codes and variable-rate codes

Adaptive Transmission in Cellular Networks: Fixed-Rate Codes With Power Control Versus Physical Layer Rateless Codes

Adaptive transmission schemes are a crucial aspect of radio design for future wireless networks. This paper studies the performance of two classes of adaptive transmission schemes in cellular downlink. One class is based on physical layer rateless codes with constant transmit power and the other uses fixed-rate codes in conjunction with power adaptation. Using a simple stochastic geometry model for the cellular downlink, the focus is to compare the adaptivity of fixed-rate codes with power adaptation to that of physical layer rateless codes only. The performance of both the rateless and fixed-rate coded adaptive transmission schemes are compared by evaluating the typical user success probability and rate achievable with the two schemes. Based on both the theoretical analysis and simulation results, it is clearly shown that the fixed-rate codes require power control to maintain good performance, whereas the physical layer rateless codes with constant power can still provide robust performance.

RESEARCH OF CHARACTERISTICS OF NOISE IMMUNITY WITH USE OF ORTHOGONAL CODING

In the digital communication systems for noise immunity's increase with the fixed code rate it is proposed to use an additional orthogonal coding developed by the authors. It is an analogue of convolutional coding over the rational numbers' field. Transmission of digital signals in Additive white Gaussian noise (AWGN) channel and fading channels is considered including a joint use of the orthogonal and correcting codes (block and convolutional). It is shown that losses in signal-to-noise ratio can be significantly reduced by use of orthogonal coding. By increase of matrices' order, on which basis orthogonal codes are constructed, the coding gain grows also. By use of the proposed by the authors orthogonal coding the required quality of communication is implemented with a smaller energy cost. The significant coding gain (up to 6,4 dB in the channels with the AWGN, up to 22,74 dB in the fading channels) provided by more effective use of energy of transmitted signals is reached without increase in complexity and cost of transmitting/receiving devices.

DEVELOPMENT OF A CLASS OF SYSTEM AND RETURN SYSTEM MATRIXES PROVIDING INCREASE IN NOISE IMMUNITY OF SPECTRALLY EFFECTIVE MODULATION SCHEMES ON BASIS OF ORTHOGONAL CODING

In work it is proposed in the digital systems of messages transmission for noise immunity's increase with the fixed code rate to use an additional coding called by the authors orthogonal. The way of a definition of orthogonal codes is presented, the synthesis algorithm of system and inverse system matrices of orthogonal codes is developed, and the main parameters of some matrices constructed by the offered algorithm are specified. Orthogonal coding as a special case of convolutional coding is defined by matrices, which elements are polynomials in the delay variable with integer coefficients. Code words are given by multiplication of an information polynomial by a system matrix, and decoding is performed by multiplication by an inverse system matrix. Basic ratios for orthogonal coding are given in article, and properties of system and inverse matrices are specified. Parameters of system and inverse system matrices assure additional gain in signal-to-noise ratio. This gain is got as a result of a more effective use of energy of transmitted signals. For transmission of one symbol energy of several symbols is accumulated.

Fixed-Rate Universal Lossy Source Coding and Model Identification: Connection with Zero-Rate Density Estimation and the Skeleton Estimator.

This work demonstrates a formal connection between density estimation with a data-rate constraint and the joint objective of fixed-rate universal lossy source coding and model identification introduced by Raginsky in 2008 (IEEE TIT, 2008, 54, 3059–3077). Using an equivalent learning formulation, we derive a necessary and sufficient condition over the class of densities for the achievability of the joint objective. The learning framework used here is the skeleton estimator, a rate-constrained learning scheme that offers achievable results for the joint coding and modeling problem by optimally adapting its learning parameters to the specific conditions of the problem. The results obtained with the skeleton estimator significantly extend the context where universal lossy source coding and model identification can be achieved, allowing for applications that move from the known case of parametric collection of densities with some smoothness and learnability conditions to the rich family of non-parametric -totally bounded densities. In addition, in the parametric case we are able to remove one of the assumptions that constrain the applicability of the original result obtaining similar performances in terms of the distortion redundancy and per-letter rate overhead.

LDPC Code Design for Distributed Storage: Balancing Repair Bandwidth, Reliability, and Storage Overhead

Distributed storage systems suffer from significant repair traffic generated due to the frequent storage node failures. This paper shows that properly designed low-density parity-check (LDPC) codes can substantially reduce the amount of required block downloads for repair thanks to the sparse nature of their factor graph representation. In particular, with a careful construction of the factor graph, both low repair-bandwidth and high reliability can be achieved for a given code rate. First, a formula for the average repair bandwidth of LDPC codes is developed. This formula is then used to establish that the minimum repair bandwidth can be achieved by forcing a regular check node degree in the factor graph. Moreover, it is shown that given a fixed code rate, the variable node degree should also be regular to yield minimum repair bandwidth, under some reasonable minimum variable node degree constraint. It is also shown that for a given repair-bandwidth requirement, LDPC codes can yield substantially higher reliability than the currently utilized Reed–Solomon codes. Our reliability analysis is based on a formulation of the general equation for the mean-time-to-data-loss (MTTDL) associated with LDPC codes. The formulation reveals that the stopping number is closely related to the MTTDL. It is further shown that LDPC codes can be designed such that a small loss of repair-bandwidth optimality may be traded for a large improvement in erasure-correction capability and thus the MTTDL.

Research and Implementation of Rateless Spinal Codes Based Massive MIMO System

The potential performance gains promised by massive multi‐input and multioutput (MIMO) rely heavily on the access to accurate channel state information (CSI), which is difficult to obtain in practice when channel coherence time is short and the number of mobile users is huge. To make the system with imperfect CSI perform well, we propose a rateless codes‐aided massive MIMO scheme, with the aim of approaching the maximum achievable rate (MAR) as well as improving the achieved rate over that based on the fixed‐rate codes. More explicitly, a recently proposed family of rateless codes, called spinal codes, are applied to massive MIMO systems, where the spinal codes bring the benefit of approximately achieving the MAR with sufficiently large encoding block size. In addition, the multilevel puncturing and dynamic block‐size allocation (MPDBA) scheme is proposed, where the block sizes are determined by user MAR to curb the average retransmission delay for successfully decoding the messages, which further enhances the system retransmission efficiency. Multilevel puncturing, which is MAR dependent, narrows the gap between the system MAR and the related achieved rate. Theoretical analysis is provided to demonstrate that spinal codes with the MPDBA can guarantee the system retransmission efficiency as well as achieved rate, which are also verified by numerical simulations. Finally, a simplified but comparable MIMO testbed with 2 transmit antennas and 2 single‐antenna users, based on NI Universal Software Radio Peripheral (USRP) and LabVIEW communication toolkits, is built up to demonstrate the effectiveness of our proposal in realistic wireless channels, which is easy to be extended to massive MIMO scenarios in future.

Utilization of Raptor Codes for OFDM-System Performance Enhancing

In this paper, we investigate the utilization of Raptor code for performance enhancement of Orthogonal Frequency Division Multiplexing (OFDM) systems. Unlike classical channel coding techniques which have fixed code rates, Raptor code is a rateless code. Raptor code has proved to have a wonderful performance over a class of wireless channels. Simulation experiments have been carried out to investigate the performance of Raptor code with OFDM. Different modulation schemes have been used in the simulation experiments such as Binary Phase Shift Keying (BPSK), Quadrature Phase Shift Keying (QPSK), and Quadrature Amplitude Modulation (QAM). The Bit Error Rate (BER) has been used as the evaluation metric for the proposed Raptor-OFDM system. Simulation results reveal a good performance of the proposed system.

Enhanced Cellular Coverage and Throughput Using Rateless Codes

Rateless codes have been shown to provide robust error correction over a wide range of binary and noisy channels. Using a stochastic geometry model, this paper studies the performance of rateless codes in the cellular downlink and compares it with the performance of fixed-rate codes. For the case of Rayleigh fading, an accurate approximation is proposed for the distribution of the packet transmission time of $K$ -bit information packets using rateless codes. The two types of channel coding schemes are compared by evaluating the typical user and per-user success probability and the rate. Based on both the analytical results and simulations, the paper shows that rateless coding provides a significant throughput gain relative to fixed-rate coding. Moreover, the benefit is not restricted to the typical user but applies to all users in the cellular network.