A closed-form scaling law for convolutional LDPC codes over the BEC

Abstract

We propose a scaling law for the error probability of convolutional LDPC ensembles when transmission takes place over the binary erasure channel. We discuss how the parameters of the scaling law are connected to fundamental quantities appearing in the asymptotic analysis of these ensembles and we verify that the predictions of the scaling law fit well with data derived from simulations over a wide range of parameters. I. INTRODUCTIONRecently, it has been proved that spatially coupled lowdensity parity-check (SC-LDPC) codes can achieve capacity over binary-input memoryless symmetric channels under belief propagation (BP) decoding [1]. SC-LDPC ensembles are constructed by coupling L (l, r)-regular LDPC codes, each one of length M bits, together with appropriate boundary conditions. When M tends to infinity and L is large, SC-LDPC codes exhibit a BP threshold arbitrarily close to the maximum-aposteriori (MAP) threshold of (l, r)-regular ensemble [1], [2]. In contrast, much less is known about the finite-length behavior of SC-LDPC codes [3]. This finite-length performance requires to relate the code parameters, namely the chain length L, the coupling pattern, and the code length n = M L, to the error probability. In this paper, we study a specific class of SC-LDPC codes, usually referred to as convolutional LPDC codes [2], when they are used for transmission over the binary erasure channel (BEC). In [4] we provided a first description of the scaling behavior of convolutional LDPC ensembles using an extensive set of simulations. However, no closed-form expression was provided to evaluate the error probability in the range of most interest, namely when we consider coupled codes above the BP threshold of (l, r)-regular LDPC code.In the present work we extend to convolutional LDPC codes the methodology proposed in [5] to analyze finite-length LDPC ensembles over the BEC. For the BEC the workings of the BP decoder can be cast in an alternative formulation, namely as peeling decoder (PD). In this formulation, any time a variable nodes gets decoded, it is removed from the graph along with all attached edges. The analysis then consists on studying the evolution of the graph. More precisely, it suffices to track the expected evolution of the graph and the variance around this expected behavior since errors in this framework correspond to large deviations from the expected behavior. If it is possible to derive expressions for the mean and the variance

A finite length performance analysis of LDPC codes constructed by connecting spatially coupled chains

“…In [8], Scaling Laws (SLs) were proposed to predict PD finite-length performance in the waterfall region, based on the graph covariance evolution around the typical path as a function of code length. A finite-length analysis of a single chain ensemble for large L is given in [9]. Here we generalize these results to any SC-LDPC ensemble defined by a T matrix.…”

“…Here we describe the computation of the graph expectations in (8) and (9). Assume the graph at iteration has DD {R j,u ( ), V u ( )} for u ∈ [1, D] and j = 1, .…”

“…A finite-length analysis of single chain ensembles has recently been addressed in [9]. In this paper, we consider PD applied to connected SC-LDPC code chains.…”

Analyzing finite-length protograph-based spatially coupled LDPC codes

“…We consider classes of SC-LDPC code ensembles based on the (l, r) regular LDPC ensembles. In this section, we present the random construction proposed in [1], [5] and its counterpart based on protographs. The random and protograph ensembles are denoted by (l, r, L) and (l, r, L) P , respectively.…”

“…However, protograph LDPC ensembles are a class of multi-edge type LDPC codes that are hard to analyze. Indeed, proofs for achieving capacity over BMS channels [1] and finite-length performance analyses [5], [6] have been proposed only for random SC-LDPC codes.…”