Threshold saturation of spatially-coupled codes on intersymbol-interference channels

Abstract

Recently, it has been observed that terminated low-density-parity-check (LDPC) convolutional codes (or spatially-coupled codes) appear to approach capacity universally across the class of binary memoryless channels. This is facilitated by the "threshold saturation" effect whereby the belief-propagation (BP) threshold of the spatially-coupled ensemble is boosted to the maximum a-posteriori (MAP) threshold of the underlying constituent ensemble.In this paper, we consider the universality of spatially-coupled codes over intersymbol-interference (ISI) channels under joint iterative decoding. More specifically, we empirically show that threshold saturation also occurs for the considered problem. This can be observed by first identifying the EXIT curve for erasure noise and the GEXIT curve for general noise that naturally obey the general area theorem. From these curves, the corresponding MAP and the BP thresholds are then numerically obtained.With the fact that regular LDPC codes can achieve the symmetric information rate (SIR) under MAP decoding, spatially-coupled codes with joint iterative decoding can universally approach the SIR of ISI channels. For the dicode erasure channel, Kudekar and Kasai recently reported very similar results based on EXIT-like curves.

A Simple Proof of Maxwell Saturation for Coupled Scalar Recursions

“…The authors observe in [9] that the phenomenon of threshold saturation extends to multiterminal problems (e.g., a noisy Slepian-Wolf problem) and can provide universality over unknown channel parameters. Threshold saturation has also been observed for the binaryadder channel [18], for intersymbol-interference channels [10], [11], [12], for message-passing decoding of code-division multiple access (CDMA) [19], [20], [21], and for iterative harddecision decoding of SC generalized LDPC codes [22]. For compressive sensing, SC measurement matrices were investigated with verification-based reconstruction in [23], shown to have excellent performance with BP decoding in [24], and proven to achieve the information-theoretic limit in [25].…”

“…Convolutional low-density parity-check (LDPC) codes, or spatially-coupled (SC) LDPC codes, were introduced in [3] and shown to have excellent belief-propagation (BP) thresholds in [4], [5], [6]. Moreover, they have recently been observed to universally approach the capacity of various channels [6], [7], [8], [9], [10], [11], [12], [13].…”

Spatially Coupled LDPC Codes Constructed From Protographs

“…Recently, it has been proven by Kudekar et al that SC-LDPC-BC ensembles are capacity achieving on memoryless binary-input symmetric-output (MBS) channels under BP decoding [21], [22]. Consequently, the principle of spatial graph coupling has attracted significant attention and has been successfully applied in many other areas of communications and signal processing, such as, for example, compressed sensing [23], [24], relay channels [25]- [28], wiretap channels [29], multiple access channels [30]- [33], broadcast channels [34], intersymbol-interference channels [35], [36], multiuser detection [37], random access [38], source coding [39], quantum codes [40], [41], and models in statistical physics [42]. Also, studies of the finite length scaling properties of SC-LDPC-BCs were performed in [43], [44] and block erasure channel performance bounds were given in [45].…”

Design of Spatially Coupled LDPC Codes Over GF $(q)$ for Windowed Decoding

“…Binary SC-LDPC code ensembles have been shown to exhibit a phenomenon called "threshold saturation" [13], [14], [15], in which, as the coupling length L grows, the BP decoding threshold saturates to the maximum a-posteriori (MAP) threshold of the corresponding uncoupled LDPC-BC ensemble, which, for the (J, K)-regular code ensembles considered in this paper, approaches channel capacity as the density of the parity-check matrix increases [16]. This threshold saturation phenomenon has been reported for a variety of code ensembles (e.g., (J, K)-regular SC-LDPC code ensembles [17], accumulate-repeat-by-4-jaggedaccumulate (AR4JA) irregular SC-LDPC code ensembles [18], bilayer SC-LDPC code ensembles [19], and MacKay-Neal and Hsu-Anastasopoulos spatially-coupled code ensembles [20]) and channel models (e.g., channels with memory [21], multiple access channels [22], intersymbolinterference channels [23], and erasure relay channels [24]), thus making SC-LDPC codes attractive candidates for practical applications requiring near-capacity performance. For a more comprehensive survey of the literature on SC-LDPC codes, refer to the introduction of [25].…”