Achievable rates and forward-backward decoding algorithms for the Gaussian relay channels under the one-code constraint

A modified joint source-channel decoding (JSCD) algorithm based on low-density parity-check (LDPC) codes and hidden Markov estimation has been presented and extended to applying over Rayleigh fading channels. The proposed algorithm utilizes the source residual redundancy lack of any priori information of the source parameters, and the modifications employ the self-checking and convergence mechanisms of LDPC codes in adjusting the error probability of the hidden Markov model (HMM). To show the validation of the proposed JSCD methods, we also give a framework for the density evolution analysis for JSCD. Simulation results show that the joint decoding scheme on Rayleigh fading channels greatly reduces the number of iterations and improves the system performance more than 0.7 dB. Moreover, the modified JSCD algorithm leads to the joint decoder to converge at least 10% faster than the unmodified method without any degradation in the decoding performance

The Schalkwijk-Kailath (SK) scheme for the AWGN channel with noise-free feedback is well-known since its coding complexity grows linearly with the coding blocklength, and it is capacity-achieving and has extremely lower decoding error probability comparing with existing excellent codes such as LDPC and Polar codes. However, extension of SK scheme to more practical scenarios is challenging since it depends heavily on the feedback success. This paper aims to investigate how to design SK-type schemes for channels with intermittent feedback and establish the relationship between the coding blocklength, the desired decoding error probability, the achievable transmission rate and secrecy level of our proposed schemes. Specifically, first, for the additive white Gaussian noise (AWGN) channel with intermittent noise-free feedback, a variation of the well-known Schalkwijk-Kailath (SK) scheme for the AWGN channel with noise-free feedback is proposed, and the corresponding achievable rate is characterized for given coding blocklength and decoding error probability. Subsequently, the proposed scheme is extended to channels with noisy intermittent feedback. To quantitatively evaluate the secrecy performance, we adopt the eavesdropper's normalized equivocation as the secrecy metric and analytically characterize the achievable secrecy level of the proposed schemes. In particular, we show that perfect weak secrecy, i.e., asymptotically vanishing information leakage rate, can be achieved under certain conditions. Numerical results show that for a given decoding error probability threshold, our proposed schemes require lower signal-to-noise ratio and significantly shorter coding blocklength comparing with LDPC code. The study of this paper may provide a way to construct efficient coding scheme for channels in the presence of intermittent feedback.

Previously, it has been shown that more feedback links may help to achieve larger finite blocklength rates of the Gaussian relay channel (GRC). However, this is only built on the idealistic assumption that the feedback channels are noise free. In this paper, we revisit the GRC with feedback by considering the situation that the feedback channels are also corrupted by additive white Gaussian noises. For given coding blocklength and decoding error probability, achievable rates of our proposed schemes are characterized. Numerical results show that comparing with those schemes of the same model without using feedback, there may still exist rate gain even the feedback channels are corrupted by noises. Besides this, we also show that comparing with the existing feedback scheme for the GRC with one noise-free feedback link, two noisy feedback links still bring rate gain for some cases.

The increasing demand for high-density and high-speed digital optical recording systems has made the development of advanced coding and signal processing techniques for fast and reliable data recovery increasingly important. In recent years, the parity-check (PC)-code-based reception technique has been widely studied for magnetic recording systems, and it is projected to be highly promising for high-density optical recording. The PC code is an inner error correction code (ECC), which can detect dominant short error events of the system, using only a few parity bits. This reduces the loss in error correction capability of the outer ECC due to random short errors and results in a simple and efficient solution to improve the overall performance. This thesis is dedicated to the design and analysis of PCcode- based recording systems to achieve higher recording capacity with low implementation complexity, for high-density blue laser disc systems. In particular, most of the key components of the PC-code-based optical recording system have been designed and optimized for different recording densities, and different proportions of white noise and media noise. During the development of advanced coding and detection techniques, it becomes necessary to investigate the system’s performance with different coding schemes and recording densities. In Chapter 2 of the thesis, we propose a generalized Braat-Hopkins model for optical recording channels, which provides a fair basis for the performance comparison of detection schemes over different coding schemes and recording densities, for channels with additive, white Gaussian noise (AWGN) and media noise. Various basic issues associated with the PC-code-based systems are investigated in Chapter 3. These include bounds for bit error rates and error event probabilities, the dominant error events at channel detector output, different PC codes, simple and efficient post-processors, the impact of error events that are split across data block boundaries and the corresponding remedy, as well as the effects of code rates and recording densities. Simulation results show that a 4-bit PC code achieves the best performance. The corresponding bit error rates (BERs) are very close to the performance bounds, at both nominal and high densities. Constrained codes, which serve as a key component in the read-write channel of data storage systems, are desired to have a high code rate with low-complexity encoder/decoder. In Chapter 4, we investigate the design of certain capacity-approaching constrained codes for optical recording systems. In particular, we derive analytically the relationship between the number of encoder states and the maximum size of these codes. We identify the minimum number of encoder states that maximizes the code rate, for any desired codeword length. The design of constrained PC codes is key to the development of PCcode- based systems, and the systematic design of efficient constrained PC codes remains a challenging problem. In Chapter 5, we propose a general and systematic code design technique for constructing capacity-approaching constrained PC codes, which can detect any type of error events of the system, with the minimum code rate loss. Compared to the rate 2/3 code without parity, the newly designed constrained 4-bit PC code achieves a performance gain of 2 dB at nominal density, and 1.5 dB at high density, at BER = 10-5. As the dominant noise for high-density optical recording systems, media noise severely degrades the performance of channel detectors and postprocessors that are designed for AWGN. In Chapter 6, we propose two novel modifications to the bit detector to combat media noise. We further develop a data-dependent post-processing scheme for error correction. Compared to the system designed without considering media noise and without PC codes, the overall performance gain of the developed scheme can be more than 11 dB at high media noise levels. In data storage systems, the ECC failure rate (EFR) serves as the ultimate measure of the data recovery performance. In Chapter 7, we develop semi-analytical approaches for estimating the EFR, and analyze EFRs for the developed PC-code-based systems, for the cases without and with interleaving. Our analysis shows that with the optimum interleaving degrees, compared to the rate 9/13 code without parity, the 2-bit PC code achieves a gain of around 0.5 dB, and the 4-bit PC code gains 0.6 dB, at high record ing density and EFR = 10-16. The thesis concludes in Chapter 8 with some suggestions for further work.

Low-density Parity check codes (LDPC) are gaining interest for high data rate application in both terrestrial and spatial communications which requires low bit error rate. This paper proposes an algorithm through which the best parity check matrix is obtained for irregular low density parity check codes, which reduces the Bit Error Rate ,Frame Error Rate and computational complexity. Tanner graph also called a bipartite graph format in an Low Density Parity Check (LDPC) code design contains many short cycles; it will produce a computational complexity in a code design process and it degrades the code performance. To overcome this effect, we proposed an new algorithm which obtains a best final parity check matrix. Throughout the paper we will give the detailed description of an efficient algorithm proposed. The complexity is studied graphically and analytically. Using proposed algorithm, cycles of length four observed in corresponding matrix are removed. Each matrix is evaluated over a noisy Additive White Gaussian Noise (AWGN) channel. Bit Error rate and Frame error rate is calculated. The results are compared with randomly generated best parity check matrix. Simulation results shows that proposed method successfully overcomes the computational complexity and meets the near Shannon limit.

Day by day the internet traffic is increasing rapidly. Challenges to improve these internet traffic requirements leads to evolution of 5G NR LDPC (New Radio Low Density Parity Check) codes which provides high coding gain and energy efficiency. BER Vs Eb/No dB curve defines the energy efficiency and coding gain of the any error correcting codes. LDPC codes which are adopted by 5G Standard are called 5G NR LDPC codes.In the design of the decoder,Message Passing Approach reduces the decoding complexity due to the sparsity of the parity check matrix. An efficient decoding algorithm is the key to the successful code. In this paper an iterative minsum approximation message passing decoding algorithm has been implemented for both BG1 and BG2. Results show that the coding gain can be improved up to 9dB.

In this paper, we present a novel class of simple soft-input soft-output (SISO) detection algorithms for transmission over phase-uncertain channels. In order to limit the complexity, the set of possible phase values (i.e., the interval [0,2pi)) is properly quantized. We then use a forward-backward (FB) algorithm in correspondence to each quantized phase value. The forward and backward metrics computed by the different FB algorithms are properly combined to make the receiver robust against the time-varying phase. We apply the proposed SISO algorithms to perform joint detection and decoding of serially concatenated low-density parity-check (LDPC) codes and differentially encoded (DE) quaternary phase shift keying (QPSK), transmitted over a phase-uncertain channel. As a validation of the effectiveness of the proposed algorithms, we show that the performance loss with respect to that of the corresponding ideal (i.e., perfectly coherent) system is limited. Besides having a limited complexity, the proposed SISO algorithms are blind, i.e., no knowledge of the channel phase statistics is required, and highly parallelizable, as desirable in high-throughput future wireless communication systems

SummaryIn this paper, we propose a general model to study the full‐duplex non‐coherent (NC) decode‐and‐forward Gaussian relay channel (RC) with energy harvesting (EH) nodes, called NC‐EH‐ , in three cases: (i) no energy transfer (ET), (ii) one‐way ET from the source (S) to the relay (R), and (iii) two‐way ET. We consider the problem of optimal power allocation in NC‐EH‐ in order to maximize the total transmitted bits from S to the destination in a given time duration. General stochastic energy arrivals at S and R with known EH times and values are assumed. In NC‐EH‐ with no ET, the complicated min‐max optimization form along with its constraints make the problem intractable. It is shown that this problem can be transformed to a solvable convex form; however, convex optimization solution does not provide the structural properties of the optimal solution. Therefore, following an alternative perspective, we investigate conditions on harvesting process of S and R where we find optimal algorithmic solution. Further, we propose some suboptimal algorithms and provide some examples, in which the algorithms are optimal. Moreover, we find a class of problems for NC‐EH‐ with one‐way ET from S to R, where the optimal algorithmic solution is devised. For NC‐EH‐ with two‐way ET, we propose general optimal algorithmic solution. Furthermore, the performance of the proposed algorithms are evaluated numerically and compared with optimal numerical convex optimization tools. Copyright © 2014 John Wiley & Sons, Ltd.

Consider the composite relay channel consisting of a set of relay channels associated to a probability measure. The current channel is a draw from its probability and in some cases arbitrary small error probability cannot be guaranteed for all channels in the set. In this paper, instead of finding the maximum achievable rate subject to a small error probability (EP) for all the channels in the set, we look at the asymptotic behavior of EP for a given rate. The notion of achievable EP is introduced as a novel performance measure for wireless relay channels. We can intuitively define it as the smallest EP that can be asymptotically achieved for a given rate. The behavior of EP is directly related to the ϵ-capacity of each channel in the set. It is shown that the behavior of EP is upper and lower bounded by the outage probability of a region which is referred to as the full error region. Then every code with a rate belonging to this region yields EP equal to one. Finally, new coding for oblivious cooperation simultaneously exploiting both Decode-and-Forward (DF) and Compress-and-Forward (CF) strategies is investigated. The Gaussian relay channel with slow fading is also discussed.

This dissertation presents a systematic exposition on finite-block-length coding theory and practice. We begin with the task of identifying the maximum achievable rates over noisy, finite-block-length constrained channels, referred to as (e, n)-capacity Cen, with e denoting target block-error probability and n the block length. We characterize how the optimum codes look like over finite-block-length constrained channels. Constructing good, short-block-length error-correcting codes defined on sparse graphs is the focus of the thesis. We propose a new, general method for constructing Tanner graphs having a large girth by progressively establishing edges or connections between symbol and check nodes in an edge-by-edge manner, called progressive edge-growth (PEG) construction. Lower bounds on the girth of PEG Tanner graphs and on the minimum distance of the resulting low-density parity-check (LDPC) codes are derived in terms of parameters of the graphs. The PEG construction attains essentially the same girth as Gallager's explicit construction for regular graphs, both of which meet or exceed an extension of the Erdos-Sachs bound. The PEG construction proves to be powerful for generating good, short-block-length binary LDPC codes. Furthermore, we show that the binary interpretation of GF(2b) codes on the cycle Tanner graph TG(2, dc), if b grows sufficiently large, can be used over the binary-input additive white Gaussian noise (AWGN) channel as good code for optimum and good code for iterative decoding. Codes on sparse graphs are often decoded iteratively by a sum-product algorithm (SPA) with low complexity. We investigate efficient digital implementations of the SPA for decoding binary LDPC codes from both the architectural and algorithmic point of view, and describe new reduced-complexity derivatives thereof. The unified treatment of decoding techniques for LDPC codes provides flexibility in selecting the appropriate design point in high-speed applications from a performance, latency, and computational complexity perspective.

Various log-likelihood-ratio-based belief-propagation (LLR-BP) decoding algorithms and their reduced-complexity derivatives for low-density parity-check (LDPC) codes are presented. Numerically accurate representations of the check-node update computation used in LLR-BP decoding are described. Furthermore, approximate representations of the decoding computations are shown to achieve a reduction in complexity by simplifying the check-node update, or symbol-node update, or both. In particular, two main approaches for simplified check-node updates are presented that are based on the so-called min-sum approximation coupled with either a normalization term or an additive offset term. Density evolution is used to analyze the performance of these decoding algorithms, to determine the optimum values of the key parameters, and to evaluate finite quantization effects. Simulation results show that these reduced-complexity decoding algorithms for LDPC codes achieve a performance very close to that of the BP algorithm. The unified treatment of decoding techniques for LDPC codes presented here provides flexibility in selecting the appropriate scheme from performance, latency, computational-complexity, and memory-requirement perspectives.

Harvesting energy from radio frequency signals along with transmitting data through them is appealing for different wireless communication scenarios, such as radio frequency identification (RFID) systems and implantable devices. In this paper, we propose a technique to design nonlinear codes for the use in such systems taking into account both energy transmission and error rate requirements. In particular, we propose using concatenation of a nonlinear trellis code (NLTC) with an outer low-density parity-check (LDPC) code. We design the NLTC based on maximization of its free distance. We give necessary and sufficient conditions for its catastrophicity; in order to avoid catastrophic codes, we connect each designed NLTC to a corresponding linear convolutional code allowing for the use of simpler conditions for verification. Furthermore, we use EXIT charts to design the outer LDPC code while fixing the inner NLTC. Via examples, we demonstrate that our designed codes operate at ∼ 0.8 dB away from the information theoretic limits, and they outperform both regular LDPC codes and optimized irregular LDPC codes for additive white Gaussian noise (AWGN) channels. In addition, we show that the proposed scheme outperforms the reference schemes of concatenating LDPC codes with nonlinear memoryless mappers and using classical linear block codes in a time switching mode. © 2016 IEEE.

The multiple description coding framework is well-suited for the transmission of images in the presence of transmission errors. However, it has been mainly explored for idealized channels, where packets are either received correctly or not at all. In more realistic scenarios, bitstreams are affected by bit-level noise and error correcting codes must be employed. In the proposed scheme, the input image is decomposed into wavelet coefficients that are encoded using lattice vector quantization for the generation of multiple descriptions. Low-density parity-check (LDPC) codes are used for the protection of each stream, while joint decoding exploits the correlation between the multiple descriptions. The proposed joint source-channel decoding scheme can provide robust transmission of images by introducing a controlled amount of redundancy that can be used for error recovery. Results are presented for image transmission over binary symmetric and Gaussian channels.

Neural prosthetic systems seek to improve the lives of severely disabled people by decoding neural activity into useful behavioral commands. These systems and their decoding algorithms are typically developed "offline," using neural activity previously gathered from a healthy animal, and the decoded movement is then compared with the true movement that accompanied the recorded neural activity. However, this offline design and testing may neglect important features of a real prosthesis, most notably the critical role of feedback control, which enables the user to adjust neural activity while using the prosthesis. We hypothesize that understanding and optimally designing high-performance decoders require an experimental platform where humans are in closed-loop with the various candidate decode systems and algorithms. It remains unexplored the extent to which the subject can, for a particular decode system, algorithm, or parameter, engage feedback and other strategies to improve decode performance. Closed-loop testing may suggest different choices than offline analyses. Here we ask if a healthy human subject, using a closed-loop neural prosthesis driven by synthetic neural activity, can inform system design. We use this online prosthesis simulator (OPS) to optimize "online" decode performance based on a key parameter of a current state-of-the-art decode algorithm, the bin width of a Kalman filter. First, we show that offline and online analyses indeed suggest different parameter choices. Previous literature and our offline analyses agree that neural activity should be analyzed in bins of 100- to 300-ms width. OPS analysis, which incorporates feedback control, suggests that much shorter bin widths (25-50 ms) yield higher decode performance. Second, we confirm this surprising finding using a closed-loop rhesus monkey prosthetic system. These findings illustrate the type of discovery made possible by the OPS, and so we hypothesize that this novel testing approach will help in the design of prosthetic systems that will translate well to human patients.

Scheduling strategy is considered an important aspect of belief-propagation (BP) decoding of low-density parity-check (LDPC) codes because it affects the decoder's convergence rate, decoding complexity and error-correction performance. In this paper, we propose two new scheduling strategies for the BP decoding of non-binary LDPC (NB-LDPC) codes. Both the strategies are devised based on the concept of check node reliability and employ a heuristically defined threshold which can adapt to the communication channel variations. As the scheduling strategies only update a subset of the check nodes in each iteration, they result in reduced iteration cost. Furthermore, since the BP performs suboptimally for finite-length LDPC codes, especially for short-length LDPC codes, by enhancing the message propagation over the Tanner Graphs of short-length NB-LDPC codes, the new scheduling strategies can even improve the error-correction performances of BP decoding. Simulation results demonstrate that the new scheduling strategies provide good performance/complexity tradeoffs.