4 research outputs found
A Simple Proof of Maxwell Saturation for Coupled Scalar Recursions
Low-density parity-check (LDPC) convolutional codes (or spatially-coupled
codes) were recently shown to approach capacity on the binary erasure channel
(BEC) and binary-input memoryless symmetric channels. The mechanism behind this
spectacular performance is now called threshold saturation via spatial
coupling. This new phenomenon is characterized by the belief-propagation
threshold of the spatially-coupled ensemble increasing to an intrinsic noise
threshold defined by the uncoupled system. In this paper, we present a simple
proof of threshold saturation that applies to a wide class of coupled scalar
recursions. Our approach is based on constructing potential functions for both
the coupled and uncoupled recursions. Our results actually show that the fixed
point of the coupled recursion is essentially determined by the minimum of the
uncoupled potential function and we refer to this phenomenon as Maxwell
saturation. A variety of examples are considered including the
density-evolution equations for: irregular LDPC codes on the BEC, irregular
low-density generator matrix codes on the BEC, a class of generalized LDPC
codes with BCH component codes, the joint iterative decoding of LDPC codes on
intersymbol-interference channels with erasure noise, and the compressed
sensing of random vectors with i.i.d. components.Comment: This article is an extended journal version of arXiv:1204.5703 and
has now been accepted to the IEEE Transactions on Information Theory. This
version adds additional explanation for some details and also corrects a
number of small typo