Convolutional low-density parity-check (LDPC) ensembles, introduced by Felstrom and Zigangirov, have excellent thresholds and these thresholds are rapidly increasing functions of the average degree. Several variations on the basic theme have been proposed to date, all of which share the good performance characteristics of convolutional LDPC ensembles. We describe the fundamental mechanism that explains why “convolutional-like” or “spatially coupled” codes perform so well. In essence, the spatial coupling of individual codes increases the belief-propagation (BP) threshold of the new ensemble to its maximum possible value, namely the maximum a posteriori (MAP) threshold of the underlying ensemble. For this reason, we call this phenomenon “threshold saturation.” This gives an entirely new way of approaching capacity. One significant advantage of this construction is that one can create capacity-approaching ensembles with an error correcting radius that is increasing in the blocklength. Although we prove the “threshold saturation” only for a specific ensemble and for the binary erasure channel (BEC), empirically the phenomenon occurs for a wide class of ensembles and channels. More generally, we conjecture that for a large range of graphical systems a similar saturation of the “dynamical” threshold occurs once individual components are coupled sufficiently strongly. This might give rise to improved algorithms and new techniques for analysis.

Threshold Saturation via Spatial Coupling: Why Convolutional LDPC Ensembles Perform So Well over the BEC

A high-throughput memory-efficient decoder architecture for low-density parity-check (LDPC) codes is proposed based on a novel turbo decoding algorithm. The architecture benefits from various optimizations performed at three levels of abstraction in system design-namely LDPC code design, decoding algorithm, and decoder architecture. First, the interconnect complexity problem of current decoder implementations is mitigated by designing architecture-aware LDPC codes having embedded structural regularity features that result in a regular and scalable message-transport network with reduced control overhead. Second, the memory overhead problem in current day decoders is reduced by more than 75% by employing a new turbo decoding algorithm for LDPC codes that removes the multiple checkto-bit message update bottleneck of the current algorithm. A new merged-schedule merge-passing algorithm is also proposed that reduces the memory overhead of the current algorithm for low to moderate-throughput decoders. Moreover, a parallel soft-input-soft-output (SISO) message update mechanism is proposed that implements the recursions of the Balh-Cocke-Jelinek-Raviv (BCJR) algorithm in terms of simple "max-quartet" operations that do not require lookup-tables and incur negligible loss in performance compared to the ideal case. Finally, an efficient programmable architecture coupled with a scalable and dynamic transport network for storing and routing messages is proposed, and a full-decoder architecture is presented. Simulations demonstrate that the proposed architecture attains a throughput of 1.92 Gb/s for a frame length of 2304 bits, and achieves savings of 89.13% and 69.83% in power consumption and silicon area over state-of-the-art, with a reduction of 60.5% in interconnect length.

http://shanbhag.ece.illinois.edu/publications/mansr-tvlsi-2003-1.pdf

High-throughput LDPC decoders

A transmission system with adjustable data rate for single-carrier coherent optical transmission is proposed, which enables high-speed transmission close to the Shannon limit. The proposed system is based on probabilistically shaped 64-QAM modulation formats. Adjustable shaping is combined with a fixed-QAM modulation and a fixed forward-error correction code to realize a system with adjustable net data rate that can operate over a large reach range. At the transmitter, an adjustable distribution matcher performs the shaping. At the receiver, an inverse distribution matcher is used. Probabilistic shaping is implemented into a coherent optical transmission system for 64-QAM at 32 Gbaud to realize adjustable operation modes for net data rates ranging from 200 to 300 Gb/s. It is experimentally demonstrated that the optical transmission of probabilistically shaped 64-QAM signals outperforms the transmission reach of regular 16-QAM and regular 64-QAM signals by more than 40% in the transmission reach.

Rate Adaptation and Reach Increase by Probabilistically Shaped 64-QAM: An Experimental Demonstration

Mobile communications have been undergoing a generational change every ten years or so. However, the time difference between the so-called “G’s” is also decreasing. While fifth-generation (5G) systems are becoming a commercial reality, there is already significant interest in systems beyond 5G, which we refer to as the sixth generation (6G) of wireless systems. In contrast to the already published papers on the topic, we take a top-down approach to 6G. More precisely, we present a holistic discussion of 6G systems beginning with lifestyle and societal changes driving the need for next-generation networks. This is followed by a discussion into the technical requirements needed to enable 6G applications, based on which we dissect key challenges and possibilities for practically realizable system solutions across all layers of the Open Systems Interconnection stack (i.e., from applications to the physical layer). Since many of the 6G applications will need access to an order-of-magnitude more spectrum, utilization of frequencies between 100 GHz and 1 THz becomes of paramount importance. As such, the 6G ecosystem will feature a diverse range of frequency bands, ranging from below 6 GHz up to 1 THz. We comprehensively characterize the limitations that must be overcome to realize working systems in these bands and provide a unique perspective on the physical and higher layer challenges relating to the design of next-generation core networks, new modulation and coding methods, novel multiple-access techniques, antenna arrays, wave propagation, radio frequency transceiver design, and real-time signal processing. We rigorously discuss the fundamental changes required in the core networks of the future, such as the redesign or significant reduction of the transport architecture that serves as a major source of latency for time-sensitive applications. This is in sharp contrast to the present hierarchical network architectures that are not suitable to realize many of the anticipated 6G services. While evaluating the strengths and weaknesses of key candidate 6G technologies, we differentiate what may be practically achievable over the next decade, relative to what is possible in theory. Keeping this in mind, we present concrete research challenges for each of the discussed system aspects, providing inspiration for what follows.

6G Wireless Systems: Vision, Requirements, Challenges, Insights, and Opportunities

An iterative decoding threshold analysis for terminated regular LDPC convolutional (LDPCC) codes is presented. Using density evolution techniques, the convergence behavior of an iterative belief propagation decoder is analyzed for the binary erasure channel and the AWGN channel with binary inputs. It is shown that for a terminated LDPCC code ensemble, the thresholds are better than for corresponding regular and irregular LDPC block codes.

/pdf/iterative-decoding-threshold-analysis-for-ldpc-convolutional-23yuqiobby.pdf

Iterative Decoding Threshold Analysis for LDPC Convolutional Codes

Potentially large storage requirements and long initial decoding delays are two practical issues related to the decoding of low-density parity-check (LDPC) convolutional codes using a continuous pipeline decoder architecture. In this paper, we propose several reduced complexity decoding strategies to lessen the storage requirements and the initial decoding delay without significant loss in performance. We also provide bit error rate comparisons of LDPC block and LDPC convolutional codes under equal processor (hardware) complexity and equal decoding delay assumptions. A partial syndrome encoder realization for LDPC convolutional codes is also proposed and analyzed. We construct terminated LDPC convolutional codes that are suitable for block transmission over a wide range of frame lengths. Simulation results show that, for terminated LDPC convolutional codes of sufficiently large memory, performance can be improved by increasing the density of the syndrome former matrix.

Implementation aspects of LDPC convolutional codes

We present a family of protograph based LDPC codes that can be derived from permutation matrix based regular (J,K) LDPC convolutional codes by termination. In the terminated protograph, all variable nodes still have degree J but some check nodes at the start and end of the protograph have degrees smaller than K. Since the fraction of these stronger nodes vanishes as the termination length L increases, we call the codes asymptotically regular. The density evolution thresholds of these protographs are better than those of regular (J, K) block codes. Interestingly, this threshold improvement gets stronger with increasing node degrees (at a fixed rate) and it does not decay as L increases. Terminated convolutional protographs can also be derived from standard irregular protographs and may exhibit a significant threshold improvement.

/pdf/approaching-capacity-with-asymptotically-regular-ldpc-codes-47opgqfxbe.pdf

Approaching capacity with asymptotically regular LDPC codes

A generalization of Gallager's low-density parity-check codes is introduced, where as component codes single-error correcting Hamming codes are used instead of single-error detecting parity-check codes. Low-density (LD) parity-check codes were first introduced by Gallager in 1963. These codes are, in combination with iterative decoding, very promising for achieving low error probabilities at a reasonable cost. Results of computer simulations for long LD codes show, that a performance close to the Shannon capacity limit can be achieved. In this work, we consider a generalized version of low-density parity-check codes, where the decoding procedure can be based on the decoding of Hamming component codes.

Kamil Sh. Zigangirov

Papers

Iterative Decoding Threshold Analysis for LDPC Convolutional Codes

Implementation aspects of LDPC convolutional codes

Approaching capacity with asymptotically regular LDPC codes

Iterative decoding of generalized low-density parity-check codes

New results on the minimum distance of repeat multiple accumulate codes