Spatially Coupled Codes and Optical Fiber Communications: An Ideal Match?

In this paper, we highlight the class of spatially coupled codes and discuss their applicability to long-haul and submarine optical communication systems. We first demonstrate how to optimize irregular spatially coupled LDPC codes for their use in optical communications with limited decoding hardware complexity and then present simulation results with an FPGA-based decoder where we show that very low error rates can be achieved and that conventional block-based LDPC codes can be outperformed. In the second part of the paper, we focus on the combination of spatially coupled LDPC codes with different demodulators and detectors, important for future systems with adaptive modulation and for varying channel characteristics. We demonstrate that SC codes can be employed as universal, channel-agnostic coding schemes.Comment: Invited paper to be presented in the special session on "Signal Processing, Coding, and Information Theory for Optical Communications" at IEEE SPAWC 201

Spatially Coupled LDPC Codes Constructed from Protographs

In this paper, we construct protograph-based spatially coupled low-density parity-check (SC-LDPC) codes by coupling together a series of L disjoint, or uncoupled, LDPC code Tanner graphs into a single coupled chain. By varying L, we obtain a flexible family of code ensembles with varying rates and frame lengths that can share the same encoding and decoding architecture for arbitrary L. We demonstrate that the resulting codes combine the best features of optimized irregular and regular codes in one design: capacity approaching iterative belief propagation (BP) decoding thresholds and linear growth of minimum distance with block length. In particular, we show that, for sufficiently large L, the BP thresholds on both the binary erasure channel (BEC) and the binary-input additive white Gaussian noise channel (AWGNC) saturate to a particular value significantly better than the BP decoding threshold and numerically indistinguishable from the optimal maximum a-posteriori (MAP) decoding threshold of the uncoupled LDPC code. When all variable nodes in the coupled chain have degree greater than two, asymptotically the error probability converges at least doubly exponentially with decoding iterations and we obtain sequences of asymptotically good LDPC codes with fast convergence rates and BP thresholds close to the Shannon limit. Further, the gap to capacity decreases as the density of the graph increases, opening up a new way to construct capacity achieving codes on memoryless binary-input symmetric-output (MBS) channels with low-complexity BP decoding.Comment: Submitted to the IEEE Transactions on Information Theor

Optimized Bit Mappings for Spatially Coupled LDPC Codes over Parallel Binary Erasure Channels

In many practical communication systems, one binary encoder/decoder pair is used to communicate over a set of parallel channels. Examples of this setup include multi-carrier transmission, rate-compatible puncturing of turbo-like codes, and bit-interleaved coded modulation (BICM). A bit mapper is commonly employed to determine how the coded bits are allocated to the channels. In this paper, we study spatially coupled low-density parity check codes over parallel channels and optimize the bit mapper using BICM as the driving example. For simplicity, the parallel bit channels that arise in BICM are replaced by independent binary erasure channels (BECs). For two parallel BECs modeled according to a 4-PAM constellation labeled by the binary reflected Gray code, the optimization results show that the decoding threshold can be improved over a uniform random bit mapper, or, alternatively, the spatial chain length of the code can be reduced for a given gap to capacity. It is also shown that for rate-loss free, circular (tail-biting) ensembles, a decoding wave effect can be initiated using only an optimized bit mapper

Improving soft FEC performance for higher-order modulations via optimized bit channel mappings

Soft forward error correction with higher-order modulations is often implemented in practice via the pragmatic bit-interleaved coded modulation paradigm, where a single binary code is mapped to a nonbinary modulation. In this paper, we study the optimization of the mapping of the coded bits to the modulation bits for a polarization-multiplexed fiber-optical system without optical inline dispersion compensation. Our focus is on protograph-based low-density parity-check (LDPC) codes which allow for an efficient hardware implementation, suitable for high-speed optical communications. The optimization is applied to the AR4JA protograph family, and further extended to protograph-based spatially coupled LDPC codes assuming a windowed decoder. Full field simulations via the split-step Fourier method are used to verify the analysis. The results show performance gains of up to 0.25 dB, which translate into a possible extension of the transmission reach by roughly up to 8%, without significantly increasing the system complexity.Comment: This paper was published in Optics Express and is made available as an electronic reprint with the permission of OSA. The paper can be found at the following URL on the OSA website: http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-22-12-1454

Finite-Length Scaling Laws for Spatially-Coupled LDPC Codes

This thesis concerns predicting the finite-length error-correcting performance of spatially-coupled low-density parity-check (SC-LDPC) code ensembles over the binary erasure channel. SC-LDPC codes are a very powerful class of codes; their use in practical communication systems, however, requires the system designer to specify a considerable number of code and decoder parameters, all of which affect both the code’s error-correcting capability and the system’s memory, energy, and latency requirements. Navigating the space of the associated trade-offs is challenging. The aim of the finite-length scaling laws proposed in this thesis is to facilitate code and decoder parameter optimization by providing a way to predict the code’s error-rate performance without resorting to Monte-Carlo simulations for each combination of code/decoder and channel parameters.First, we tackle the problem of predicting the frame, bit, and block error rate of SC-LDPC code ensembles over the binary erasure channel under both belief propagation (BP) decoding and sliding window decoding when the maximum number of decoding iterations is unlimited. The scaling laws we develop provide very accurate predictions of the error rates.Second, we derive a scaling law to accurately predict the bit and block error rate of SC-LDPC code ensembles with doping, a technique relevant for streaming applications for limiting the inherent rate loss of SC-LDPC codes. We then use the derived scaling law for code parameter optimization and show that doping can offer a way to achieve better transmission rates for the same target bit error rate than is possible without doping.Last, we address the most challenging (and most practically relevant) case where the maximum number of decoding iterations is limited, both for BP and sliding window decoding. The resulting predictions are again very accurate.Together, these contributions make finite-length SC-LDPC code and decoder parameter optimization via finite-length scaling laws feasible for the design of practical communication systems

ALOHA Random Access that Operates as a Rateless Code

Various applications of wireless Machine-to-Machine (M2M) communications have rekindled the research interest in random access protocols, suitable to support a large number of connected devices. Slotted ALOHA and its derivatives represent a simple solution for distributed random access in wireless networks. Recently, a framed version of slotted ALOHA gained renewed interest due to the incorporation of successive interference cancellation (SIC) in the scheme, which resulted in substantially higher throughputs. Based on similar principles and inspired by the rateless coding paradigm, a frameless approach for distributed random access in slotted ALOHA framework is described in this paper. The proposed approach shares an operational analogy with rateless coding, expressed both through the user access strategy and the adaptive length of the contention period, with the objective to end the contention when the instantaneous throughput is maximized. The paper presents the related analysis, providing heuristic criteria for terminating the contention period and showing that very high throughputs can be achieved, even for a low number for contending users. The demonstrated results potentially have more direct practical implications compared to the approaches for coded random access that lead to high throughputs only asymptotically.Comment: Revised version submitted to IEEE Transactions on Communication

Analysis and Design of Partially Information- and Partially Parity-Coupled Turbo Codes

In this paper, we study a class of spatially coupled turbo codes, namely partially information- and partially parity-coupled turbo codes. This class of codes enjoy several advantages such as flexible code rate adjustment by varying the coupling ratio and the encoding and decoding architectures of the underlying component codes can remain unchanged. For this work, we first provide the construction methods for partially coupled turbo codes with coupling memory $m$ and study the corresponding graph models. We then derive the density evolution equations for the corresponding ensembles on the binary erasure channel to precisely compute their iterative decoding thresholds. Rate-compatible designs and their decoding thresholds are also provided, where the coupling and puncturing ratios are jointly optimized to achieve the largest decoding threshold for a given target code rate. Our results show that for a wide range of code rates, the proposed codes attain close-to-capacity performance and the decoding performance improves with increasing the coupling memory. In particular, the proposed partially parity-coupled turbo codes have thresholds within 0.0002 of the BEC capacity for rates ranging from $1/3$ to $9/10$, yielding an attractive way for constructing rate-compatible capacity-approaching channel codes.Comment: 15 pages, 13 figures. Accepted for publication in IEEE Transactions on Communication