4,709 research outputs found
Degree Optimization and Stability Condition for the Min-Sum Decoder
The min-sum (MS) algorithm is arguably the second most fundamental algorithm
in the realm of message passing due to its optimality (for a tree code) with
respect to the {\em block error} probability \cite{Wiberg}. There also seems to
be a fundamental relationship of MS decoding with the linear programming
decoder \cite{Koetter}. Despite its importance, its fundamental properties have
not nearly been studied as well as those of the sum-product (also known as BP)
algorithm.
We address two questions related to the MS rule. First, we characterize the
stability condition under MS decoding. It turns out to be essentially the same
condition as under BP decoding. Second, we perform a degree distribution
optimization. Contrary to the case of BP decoding, under MS decoding the
thresholds of the best degree distributions for standard irregular LDPC
ensembles are significantly bounded away from the Shannon threshold. More
precisely, on the AWGN channel, for the best codes that we find, the gap to
capacity is 1dB for a rate 0.3 code and it is 0.4dB when the rate is 0.9 (the
gap decreases monotonically as we increase the rate).
We also used the optimization procedure to design codes for modified MS
algorithm where the output of the check node is scaled by a constant
. For , we observed that the gap to capacity was
lesser for the modified MS algorithm when compared with the MS algorithm.
However, it was still quite large, varying from 0.75 dB to 0.2 dB for rates
between 0.3 and 0.9.
We conclude by posing what we consider to be the most important open
questions related to the MS algorithm.Comment: submitted to ITW 0
The Effect of Saturation on Belief Propagation Decoding of LDPC Codes
We consider the effect of LLR saturation on belief propagation decoding of
low-density parity-check codes. Saturation occurs universally in practice and
is known to have a significant effect on error floor performance. Our focus is
on threshold analysis and stability of density evolution.
We analyze the decoder for certain low-density parity-check code ensembles
and show that belief propagation decoding generally degrades gracefully with
saturation. Stability of density evolution is, on the other hand, rather
strongly affected by saturation and the asymptotic qualitative effect of
saturation is similar to reduction of variable node degree by one.Comment: Submitted to ISIT. Longer version to be submitted to IT Transactions
in preparatio
Protograph-Based LDPC Code Design for Ternary Message Passing Decoding
A ternary message passing (TMP) decoding algorithm for low-density
parity-check codes is developed. All messages exchanged between variable and
check nodes have a ternary alphabet, and the variable nodes exploit soft
information from the channel. A density evolution analysis is developed for
unstructured and protograph-based ensembles. For unstructured ensembles the
stability condition is derived. Optimized ensembles for TMP decoding show
asymptotic gains of up to 0.6 dB with respect to ensembles optimized for binary
message passing decoding. Finite length simulations of codes from TMP-optimized
ensembles show gains of up to 0.5 dB under TMP compared to protograph-based
codes designed for unquantized belief propagation decoding
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