35 research outputs found
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
Exact Free Distance and Trapping Set Growth Rates for LDPC Convolutional Codes
Ensembles of (J,K)-regular low-density parity-check convolutional (LDPCC)
codes are known to be asymptotically good, in the sense that the minimum free
distance grows linearly with the constraint length. In this paper, we use a
protograph-based analysis of terminated LDPCC codes to obtain an upper bound on
the free distance growth rate of ensembles of periodically time-varying LDPCC
codes. This bound is compared to a lower bound and evaluated numerically. It is
found that, for a sufficiently large period, the bounds coincide. This approach
is then extended to obtain bounds on the trapping set numbers, which define the
size of the smallest, non-empty trapping sets, for these asymptotically good,
periodically time-varying LDPCC code ensembles.Comment: To be presented at the 2011 IEEE International Symposium on
Information Theor
On the Minimum Distance of Generalized Spatially Coupled LDPC Codes
Families of generalized spatially-coupled low-density parity-check (GSC-LDPC)
code ensembles can be formed by terminating protograph-based generalized LDPC
convolutional (GLDPCC) codes. It has previously been shown that ensembles of
GSC-LDPC codes constructed from a protograph have better iterative decoding
thresholds than their block code counterparts, and that, for large termination
lengths, their thresholds coincide with the maximum a-posteriori (MAP) decoding
threshold of the underlying generalized LDPC block code ensemble. Here we show
that, in addition to their excellent iterative decoding thresholds, ensembles
of GSC-LDPC codes are asymptotically good and have large minimum distance
growth rates.Comment: Submitted to the IEEE International Symposium on Information Theory
201
Asymptotically Good LDPC Convolutional Codes Based on Protographs
LDPC convolutional codes have been shown to be capable of achieving the same
capacity-approaching performance as LDPC block codes with iterative
message-passing decoding. In this paper, asymptotic methods are used to
calculate a lower bound on the free distance for several ensembles of
asymptotically good protograph-based LDPC convolutional codes. Further, we show
that the free distance to constraint length ratio of the LDPC convolutional
codes exceeds the minimum distance to block length ratio of corresponding LDPC
block codes.Comment: Proceedings of the 2008 IEEE International Symposium on Information
Theory, Toronto, ON, Canada, July 6 - 11, 200
Wave-like Decoding of Tail-biting Spatially Coupled LDPC Codes Through Iterative Demapping
For finite coupling lengths, terminated spatially coupled low-density
parity-check (SC-LDPC) codes show a non-negligible rate-loss. In this paper, we
investigate if this rate loss can be mitigated by tail-biting SC-LDPC codes in
conjunction with iterative demapping of higher order modulation formats.
Therefore, we examine the BP threshold of different coupled and uncoupled
ensembles. A comparison between the decoding thresholds approximated by EXIT
charts and the density evolution results of the coupled and uncoupled ensemble
is given. We investigate the effect and potential of different labelings for
such a set-up using per-bit EXIT curves, and exemplify the method for a 16-QAM
system, e.g., using set partitioning labelings. A hybrid mapping is proposed,
where different sub-blocks use different labelings in order to further optimize
the decoding thresholds of tail-biting codes, while the computational
complexity overhead through iterative demapping remains small.Comment: presentat at the International Symposium on Turbo Codes & Iterative
Information Processing (ISTC), Brest, Sept. 201
Wave-like Decoding of Tail-biting Spatially Coupled LDPC Codes Through Iterative Demapping
For finite coupling lengths, terminated spatially coupled low-density
parity-check (SC-LDPC) codes show a non-negligible rate-loss. In this paper, we
investigate if this rate loss can be mitigated by tail-biting SC-LDPC codes in
conjunction with iterative demapping of higher order modulation formats.
Therefore, we examine the BP threshold of different coupled and uncoupled
ensembles. A comparison between the decoding thresholds approximated by EXIT
charts and the density evolution results of the coupled and uncoupled ensemble
is given. We investigate the effect and potential of different labelings for
such a set-up using per-bit EXIT curves, and exemplify the method for a 16-QAM
system, e.g., using set partitioning labelings. A hybrid mapping is proposed,
where different sub-blocks use different labelings in order to further optimize
the decoding thresholds of tail-biting codes, while the computational
complexity overhead through iterative demapping remains small.Comment: presentat at the International Symposium on Turbo Codes & Iterative
Information Processing (ISTC), Brest, Sept. 201
Mathematical approach to channel codes with a diagonal matrix structure
Digital communications have now become a fundamental part of modern society. In communications,
channel coding is an effective way to reduce the information rate down to channel
capacity so that the information can be transmitted reliably through the channel. This thesis is
devoted to studying the mathematical theory and analysis of channel codes that possess a useful
diagonal structure in the parity-check and generator matrices. The first aspect of these codes
that is studied is the ability to describe the parity-check matrix of a code with sliding diagonal
structure using polynomials. Using this framework, an efficient new method is proposed to obtain
a generator matrix G from certain types of parity-check matrices with a so-called defective
cyclic block structure. By the nature of this method, G can also be completely described by a
polynomial, which leads to efficient encoder design using shift registers. In addition, there is no
need for the matrices to be in systematic form, thus avoiding the need for Gaussian elimination.
Following this work, we proceed to explore some of the properties of diagonally structured lowdensity
parity-check (LDPC) convolutional codes. LDPC convolutional codes have been shown
to be capable of achieving the same capacity-approaching performance as LDPC block codes
with iterative message-passing decoding. The first crucial property studied is the minimum
free distance of LDPC convolutional code ensembles, an important parameter contributing to
the error-correcting capability of the code. Here, asymptotic methods are used to form lower
bounds on the ratio of the free distance to constraint length for several ensembles of asymptotically
good, protograph-based LDPC convolutional codes. Further, it is shown that this ratio
of free distance to constraint length for such LDPC convolutional codes exceeds the ratio of
minimum distance to block length for corresponding LDPC block codes.
Another interesting property of these codes is the way in which the structure affects the performance
in the infamous error floor (which occurs at high signal to noise ratio) of the bit error
rate curve. It has been suggested that ânear-codewordsâ may be a significant factor affecting
decoding failures of LDPC codes over an additive white Gaussian noise (AWGN) channel.
A near-codeword is a sequence that satisfies almost all of the check equations. These nearcodewords
can be associated with so-called âtrapping setsâ that exist in the Tanner graph of a
code. In the final major contribution of the thesis, trapping sets of protograph-based LDPC convolutional
codes are analysed. Here, asymptotic methods are used to calculate a lower bound
for the trapping set growth rates for several ensembles of asymptotically good protograph-based
LDPC convolutional codes. This value can be used to predict where the error floor will occur
for these codes under iterative message-passing decoding