186 research outputs found
On Universal Properties of Capacity-Approaching LDPC Ensembles
This paper is focused on the derivation of some universal properties of
capacity-approaching low-density parity-check (LDPC) code ensembles whose
transmission takes place over memoryless binary-input output-symmetric (MBIOS)
channels. Properties of the degree distributions, graphical complexity and the
number of fundamental cycles in the bipartite graphs are considered via the
derivation of information-theoretic bounds. These bounds are expressed in terms
of the target block/ bit error probability and the gap (in rate) to capacity.
Most of the bounds are general for any decoding algorithm, and some others are
proved under belief propagation (BP) decoding. Proving these bounds under a
certain decoding algorithm, validates them automatically also under any
sub-optimal decoding algorithm. A proper modification of these bounds makes
them universal for the set of all MBIOS channels which exhibit a given
capacity. Bounds on the degree distributions and graphical complexity apply to
finite-length LDPC codes and to the asymptotic case of an infinite block
length. The bounds are compared with capacity-approaching LDPC code ensembles
under BP decoding, and they are shown to be informative and are easy to
calculate. Finally, some interesting open problems are considered.Comment: Published in the IEEE Trans. on Information Theory, vol. 55, no. 7,
pp. 2956 - 2990, July 200
Spatially Coupled LDPC Codes Constructed from Protographs
In this paper, we construct protograph-based spatially coupled low-density
parity-check (SC-LDPC) codes by coupling together a series of L disjoint, or
uncoupled, LDPC code Tanner graphs into a single coupled chain. By varying L,
we obtain a flexible family of code ensembles with varying rates and frame
lengths that can share the same encoding and decoding architecture for
arbitrary L. We demonstrate that the resulting codes combine the best features
of optimized irregular and regular codes in one design: capacity approaching
iterative belief propagation (BP) decoding thresholds and linear growth of
minimum distance with block length. In particular, we show that, for
sufficiently large L, the BP thresholds on both the binary erasure channel
(BEC) and the binary-input additive white Gaussian noise channel (AWGNC)
saturate to a particular value significantly better than the BP decoding
threshold and numerically indistinguishable from the optimal maximum
a-posteriori (MAP) decoding threshold of the uncoupled LDPC code. When all
variable nodes in the coupled chain have degree greater than two,
asymptotically the error probability converges at least doubly exponentially
with decoding iterations and we obtain sequences of asymptotically good LDPC
codes with fast convergence rates and BP thresholds close to the Shannon limit.
Further, the gap to capacity decreases as the density of the graph increases,
opening up a new way to construct capacity achieving codes on memoryless
binary-input symmetric-output (MBS) channels with low-complexity BP decoding.Comment: Submitted to the IEEE Transactions on Information Theor
Density Evolution for Asymmetric Memoryless Channels
Density evolution is one of the most powerful analytical tools for
low-density parity-check (LDPC) codes and graph codes with message passing
decoding algorithms. With channel symmetry as one of its fundamental
assumptions, density evolution (DE) has been widely and successfully applied to
different channels, including binary erasure channels, binary symmetric
channels, binary additive white Gaussian noise channels, etc. This paper
generalizes density evolution for non-symmetric memoryless channels, which in
turn broadens the applications to general memoryless channels, e.g. z-channels,
composite white Gaussian noise channels, etc. The central theorem underpinning
this generalization is the convergence to perfect projection for any fixed size
supporting tree. A new iterative formula of the same complexity is then
presented and the necessary theorems for the performance concentration theorems
are developed. Several properties of the new density evolution method are
explored, including stability results for general asymmetric memoryless
channels. Simulations, code optimizations, and possible new applications
suggested by this new density evolution method are also provided. This result
is also used to prove the typicality of linear LDPC codes among the coset code
ensemble when the minimum check node degree is sufficiently large. It is shown
that the convergence to perfect projection is essential to the belief
propagation algorithm even when only symmetric channels are considered. Hence
the proof of the convergence to perfect projection serves also as a completion
of the theory of classical density evolution for symmetric memoryless channels.Comment: To appear in the IEEE Transactions on Information Theor
Bilayer Low-Density Parity-Check Codes for Decode-and-Forward in Relay Channels
This paper describes an efficient implementation of binning for the relay
channel using low-density parity-check (LDPC) codes. We devise bilayer LDPC
codes to approach the theoretically promised rate of the decode-and-forward
relaying strategy by incorporating relay-generated information bits in
specially designed bilayer graphical code structures. While conventional LDPC
codes are sensitively tuned to operate efficiently at a certain channel
parameter, the proposed bilayer LDPC codes are capable of working at two
different channel parameters and two different rates: that at the relay and at
the destination. To analyze the performance of bilayer LDPC codes, bilayer
density evolution is devised as an extension of the standard density evolution
algorithm. Based on bilayer density evolution, a design methodology is
developed for the bilayer codes in which the degree distribution is iteratively
improved using linear programming. Further, in order to approach the
theoretical decode-and-forward rate for a wide range of channel parameters,
this paper proposes two different forms bilayer codes, the bilayer-expurgated
and bilayer-lengthened codes. It is demonstrated that a properly designed
bilayer LDPC code can achieve an asymptotic infinite-length threshold within
0.24 dB gap to the Shannon limits of two different channels simultaneously for
a wide range of channel parameters. By practical code construction,
finite-length bilayer codes are shown to be able to approach within a 0.6 dB
gap to the theoretical decode-and-forward rate of the relay channel at a block
length of and a bit-error probability (BER) of . Finally, it is
demonstrated that a generalized version of the proposed bilayer code
construction is applicable to relay networks with multiple relays.Comment: Submitted to IEEE Trans. Info. Theor
New Codes on Graphs Constructed by Connecting Spatially Coupled Chains
A novel code construction based on spatially coupled low-density parity-check
(SC-LDPC) codes is presented. The proposed code ensembles are described by
protographs, comprised of several protograph-based chains characterizing
individual SC-LDPC codes. We demonstrate that code ensembles obtained by
connecting appropriately chosen SC-LDPC code chains at specific points have
improved iterative decoding thresholds compared to those of single SC-LDPC
coupled chains. In addition, it is shown that the improved decoding properties
of the connected ensembles result in reduced decoding complexity required to
achieve a specific bit error probability. The constructed ensembles are also
asymptotically good, in the sense that the minimum distance grows linearly with
the block length. Finally, we show that the improved asymptotic properties of
the connected chain ensembles also translate into improved finite length
performance.Comment: Submitted to IEEE Transactions on Information Theor
The Generalized Area Theorem and Some of its Consequences
There is a fundamental relationship between belief propagation and maximum a
posteriori decoding. The case of transmission over the binary erasure channel
was investigated in detail in a companion paper. This paper investigates the
extension to general memoryless channels (paying special attention to the
binary case). An area theorem for transmission over general memoryless channels
is introduced and some of its many consequences are discussed. We show that
this area theorem gives rise to an upper-bound on the maximum a posteriori
threshold for sparse graph codes. In situations where this bound is tight, the
extrinsic soft bit estimates delivered by the belief propagation decoder
coincide with the correct a posteriori probabilities above the maximum a
posteriori threshold. More generally, it is conjectured that the fundamental
relationship between the maximum a posteriori and the belief propagation
decoder which was observed for transmission over the binary erasure channel
carries over to the general case. We finally demonstrate that in order for the
design rate of an ensemble to approach the capacity under belief propagation
decoding the component codes have to be perfectly matched, a statement which is
well known for the special case of transmission over the binary erasure
channel.Comment: 27 pages, 46 ps figure
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