Spatially Coupled LDPC codes affected by a single random burst of erasures

Abstract

Abstract-Spatially-Coupled LDPC (SC-LDPC) ensembles achieve the capacity of binary memoryless channels (BMS), asymptotically, under belief-propagation (BP) decoding. In this paper, we study the BP decoding of these code ensembles over a BMS channel and in the presence of a single random burst of erasures. We show that in the limit of code length, codewords can be recovered successfully if the length of the burst is smaller than some maximum recoverable burst length. We observe that the maximum recoverable burst length is practically the same if the transmission takes place over binary erasure channel or over binary additive white Gaussian channel with the same capacity. Analyzing the stopping sets, we also estimate the decoding failure probability (the error floor) when the code length is finite.

Finite-Length Analysis of Spatially-Coupled Regular LDPC Ensembles on Burst-Erasure Channels

“…III. ERROR ANALYSIS ON THE SPBC As highlighted above, we empirically observed in [2] that the worst burst scenario is when the starting position of the burst is the first bit of a spatial position in the codeword. Hence, we first analyze the performance of SC-LDPC code ensembles on the SPBC in this section and later generalize to the RBC.…”

“…To illustrate the accuracy of our bounds, we carried out Monte-Carlo simulations where we randomly erased a complete spatial position (i.e., all M VNs) from the middle of the graph (to avoid boundary effects) for each transmitted codeword. At the receiver we performed BP decoding and (3) and (2). We observe that the bound indeed is very accurate for M large enough ( 180), since large-size stopping sets (larger than 2) vanish.…”

“…Note that z 0 is arbitrary 1 . We define the normalized quantities s S M and β b M , so that 0 ≤ s ≤ 1 and 0 ≤ β ≤ L. Using density evolution in the limit of M , we showed numerically in [2] that for any fixed s, there is a maximum normalized β(s) recoverable with an arbitrarily small probability of decoding failure. Then, a burst of length βM and random starting position is recoverable in the limit of M if β < β max = min 0≤s<1 β(s).…”

“…Later, we will extend our results to arbitrary values of S and b and then, to the case of occurring more than a single-burst. We refer the interested reader to [2] for the details of density evolution and focus in this paper on the error floor due to the occurrence of a random burst of length b < β max M when M is finite.…”

“…Additionally, the burst erasure affects complete SPs. In [2], we considered a more general model. First we allowed the burst erasure to take on any length and assumed that its starting position can occur anywhere in the codeword.…”