A Phenomenological Study on Threshold Improvement via Spatial Coupling

Abstract

Kudekar et al. proved an interesting result in low-density parity-check (LDPC) convolutional codes: The belief-propagation (BP) threshold is boosted to the maximum-a-posteriori (MAP) threshold by spatial coupling. Furthermore, the authors showed that the BP threshold for code-division multiple-access (CDMA) systems is improved up to the optimal one via spatial coupling. In this letter, a phenomenological model for elucidating the essence of these phenomenon, called threshold improvement, is proposed. The main result implies that threshold improvement occurs for spatially-coupled general graphical models.Comment: re-submitted to IEICE Trans. Fundamental

Spatially Coupled Ensembles Universally Achieve Capacity Under Belief Propagation

“…This suggests that it might be possible to study difficult theoretical problems in this area, like the existence of the static threshold, by studying the dynamical threshold of a chain of coupled models, perhaps an easier problem. Further spatially-coupled models were considered by Takeuchi and Tanaka [95].…”

Threshold Saturation for Spatially Coupled LDPC and LDGM Codes on BMS Channels

“…Recently, a simple approach, based on potential functions, is used in [24], [25] to prove that the BP threshold of spatiallycoupled irregular LDPC ensembles over a BEC saturates to the conjectured MAP threshold (known as the Maxwell threshold) of the underlying irregular ensembles. This technique was motivated by [26] and is also related to the continuum approach to density evolution (DE) in which potential functions are used to prove threshold saturation for compressed sensing [23].…”

A generalization of threshold saturation: application to spatially coupled BICM-ID

“…Hassani et al [13] presented an intuitive explanation of this statement based on classical mechanics. See [4], [14] for a rigorous proof based on the intuition.…”

“…for all i ≥ I, with u(x) = lim i→∞ u(x, i) denoting the stationary solution to the integral systems (24) and (25). With this number I of iterations we use the triangle inequality for the left-hand side (LHS) of(14) to obtain 1 L l∈L |u l (i) −ũ(x l )| < 1 L l∈L |u l (I) − u(x l , I)| + 1 L l∈L |u(x l ) −ũ(x l )| + 2ǫ,(36)for all i ≥ I. From the second property of Lemma 2, we find that the first term on the upper bound (36) tends to zero in the continuum limit W = αL → ∞.…”