Threshold saturation via spatial coupling: Why convolutional LDPC ensembles perform so well over the BEC

Abstract

Abstract-Convolutional LDPC ensembles, introduced by Felström 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 which explains why "convolutional-like" or "spatially coupled" codes perform so well. In essence, the spatial coupling of the individual code structure has the effect of increasing the belief-propagation threshold of the new ensemble to its maximum possible value, namely the maximum-a-posteriori 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 such a construction is that one can create capacity-approaching ensembles with an error correcting radius which is increasing in the blocklength. Our proof makes use of the area theorem of the belief-propagation EXIT curve and the connection between the maximum-a-posteriori and beliefpropagation threshold recently pointed out by Méasson, Montanari, Richardson, and Urbanke.Although we prove the connection between the maximuma-posteriori and the belief-propagation threshold only for a very specific ensemble and only for the binary erasure channel, empirically a threshold saturation 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 as well as to new techniques for analysis.

Rigorous dynamics of expectation-propagation-based signal recovery from unitarily invariant measurements

“…Gaussian measurements, when the compression rate is larger than the BP threshold. Spatially coupled measurement matrices are required for achieving the optimal performance in the whole regime [7], [13]- [15]. However, it is recognized that AMP fails to converge when the i.i.d.…”

Bilayer LDPC convolutional codes for half-duplex relay channels

“…For each variable node, M t is uniformly and independently chosen from all possible types. It has been stated in [9] …”

“…We assume that each of the l edges of a variable node at time t uniformly and independently connects to the check nodes in the time range [t, ..., t + w]. More precisely, for each variable node at time t, one can define a type M t 1 [9] which is a w-tuple of non-negative integers, M t = (m t,t , ..., m t,t+j , ..., m t,t+w ), j ∈ [0, w], and j m t,t+j = l. The element m t,t+j indicates that there are m t,t+j edges connecting the designated variable node at time t and the check nodes at time t + j. For each variable node, M t is uniformly and independently chosen from all possible types.…”

“…It has been shown in [9] that the {l, r, L, w} ensemble with infinite M has the following properties in a binary erasure channel: for the rate of the code R…”

“…Then the idea was further developed in, e.g., [7], [8]. Recently, it has been proven This work was supported in part by the European Community's Seventh Framework Programme under grant agreement no 257626 (ACROPOLIS) and the Swedish Foundation for Strategic Research. analytically in [9] that the belief-propagation (BP) decoding threshold of an LDPC convolutional code achieves the optimal maximum a posteriori probability (MAP) threshold of the corresponding LDPC block code with the same variable and check degrees. This code in turn approaches the capacity as the node degrees increase.…”