Braided Convolutional Codes -- A Class of Spatially Coupled Turbo-Like Codes

In this paper, we investigate the impact of spatial coupling on the thresholds of turbo-like codes. Parallel concatenated and serially concatenated convolutional codes as well as braided convolutional codes (BCCs) are compared by means of an exact density evolution (DE) analysis for the binary erasure channel (BEC). We propose two extensions of the original BCC ensemble to improve its threshold and demonstrate that their BP thresholds approach the maximum-a-posteriori (MAP) threshold of the uncoupled ensemble. A comparison of the different ensembles shows that parallel concatenated ensembles can be outperformed by both serially concatenated and BCC ensembles, although they have the best BP thresholds in the uncoupled case.Comment: Invited paper, International Conference on Signal Processing and Communications, SPCOM 2014, Bangalore, India, July 22-25, 201

Spatially-Coupled Random Access on Graphs

In this paper we investigate the effect of spatial coupling applied to the recently-proposed coded slotted ALOHA (CSA) random access protocol. Thanks to the bridge between the graphical model describing the iterative interference cancelation process of CSA over the random access frame and the erasure recovery process of low-density parity-check (LDPC) codes over the binary erasure channel (BEC), we propose an access protocol which is inspired by the convolutional LDPC code construction. The proposed protocol exploits the terminations of its graphical model to achieve the spatial coupling effect, attaining performance close to the theoretical limits of CSA. As for the convolutional LDPC code case, large iterative decoding thresholds are obtained by simply increasing the density of the graph. We show that the threshold saturation effect takes place by defining a suitable counterpart of the maximum-a-posteriori decoding threshold of spatially-coupled LDPC code ensembles. In the asymptotic setting, the proposed scheme allows sustaining a traffic close to 1 [packets/slot].Comment: To be presented at IEEE ISIT 2012, Bosto

Spatially Coupled Turbo Codes

In this paper, we introduce the concept of spatially coupled turbo codes (SC-TCs), as the turbo codes counterpart of spatially coupled low-density parity-check codes. We describe spatial coupling for both Berrou et al. and Benedetto et al. parallel and serially concatenated codes. For the binary erasure channel, we derive the exact density evolution (DE) equations of SC-TCs by using the method proposed by Kurkoski et al. to compute the decoding erasure probability of convolutional encoders. Using DE, we then analyze the asymptotic behavior of SC-TCs. We observe that the belief propagation (BP) threshold of SC-TCs improves with respect to that of the uncoupled ensemble and approaches its maximum a posteriori threshold. This phenomenon is especially significant for serially concatenated codes, whose uncoupled ensemble suffers from a poor BP threshold.Comment: in Proc. 8th International Symposium on Turbo Codes & Iterative Information Processing 2014, Bremen, Germany, August 2014. To appear. (The PCC ensemble is changed with respect to the one in the previous version of the paper. However, it gives identical thresholds

Threshold Saturation for Spatially Coupled Turbo-like Codes over the Binary Erasure Channel

In this paper we prove threshold saturation for spatially coupled turbo codes (SC-TCs) and braided convolutional codes (BCCs) over the binary erasure channel. We introduce a compact graph representation for the ensembles of SC-TC and BCC codes which simplifies their description and the analysis of the message passing decoding. We demonstrate that by few assumptions in the ensembles of these codes, it is possible to rewrite their vector recursions in a form which places these ensembles under the category of scalar admissible systems. This allows us to define potential functions and prove threshold saturation using the proof technique introduced by Yedla et al..Comment: 5 pages, 3figure

A Unified Ensemble of Concatenated Convolutional Codes

We introduce a unified ensemble for turbo-like codes (TCs) that contains the four main classes of TCs: parallel concatenated codes, serially concatenated codes, hybrid concatenated codes, and braided convolutional codes. We show that for each of the original classes of TCs, it is possible to find an equivalent ensemble by proper selection of the design parameters in the unified ensemble. We also derive the density evolution (DE) equations for this ensemble over the binary erasure channel. The thresholds obtained from the DE indicate that the TC ensembles from the unified ensemble have similar asymptotic behavior to the original TC ensembles

Improving Short-Length LDPC Codes with a CRC and Iterative Ordered Statistic Decoding

We present a CRC-aided decodingscheme of LDPC codes that can outperform the underlying LDPC code underordered statistic decoding (OSD). In this scheme, the CRC is usedjointly with the LDPC code to construct a candidate list, insteadof conventionally being regarded as a detection code to prunethe list generated by the LDPC code alone. As an example weconsider a (128,64) 5G LDPC code with BP decoding, which we canoutperform by 2dB using a (128,72) LDPC code in combinationwith a 8-bit CRC under OSD order of 3. The CRC-aided decoding scheme also achieves a better performance than the conventional one where CRC is used to prune the list. A manageable complexity can be achievedwith iterative reliability based OSD, which is demonstrated toperform well with a small OSD order

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

Spatially Coupled Turbo Codes: Principles and Finite Length Performance

In this paper, we give an overview of spatially coupled turbo codes (SC-TCs), the spatial coupling of parallel and serially concatenated convolutional codes, recently introduced by the authors. For presentation purposes, we focus on spatially coupled serially concatenated codes (SC-SCCs). We review the main principles of SC-TCs and discuss their exact density evolution (DE) analysis on the binary erasure channel. We also consider the construction of a family of rate-compatible SC-SCCs with simple 4-state component encoders. For all considered code rates, threshold saturation of the belief propagation (BP) to the maximum a posteriori threshold of the uncoupled ensemble is demonstrated, and it is shown that the BP threshold approaches the Shannon limit as the coupling memory increases. Finally we give some simulation results for finite lengths.Comment: Invited paper, IEEE Int. Symp. Wireless Communications Systems (ISWCS), Aug. 201

On the Minimum Distance of Generalized Spatially Coupled LDPC Codes

Families of generalized spatially-coupled low-density parity-check (GSC-LDPC) code ensembles can be formed by terminating protograph-based generalized LDPC convolutional (GLDPCC) codes. It has previously been shown that ensembles of GSC-LDPC codes constructed from a protograph have better iterative decoding thresholds than their block code counterparts, and that, for large termination lengths, their thresholds coincide with the maximum a-posteriori (MAP) decoding threshold of the underlying generalized LDPC block code ensemble. Here we show that, in addition to their excellent iterative decoding thresholds, ensembles of GSC-LDPC codes are asymptotically good and have large minimum distance growth rates.Comment: Submitted to the IEEE International Symposium on Information Theory 201

Terminated LDPC Convolutional Codes with Thresholds Close to Capacity

An ensemble of LDPC convolutional codes with parity-check matrices composed of permutation matrices is considered. The convergence of the iterative belief propagation based decoder for terminated convolutional codes in the ensemble is analyzed for binary-input output-symmetric memoryless channels using density evolution techniques. We observe that the structured irregularity in the Tanner graph of the codes leads to significantly better thresholds when compared to corresponding LDPC block codes.Comment: To appear in the proceedings of the 2005 IEEE International Symposium on Information Theory, Adelaide, Australia, September 4-9, 200