477 research outputs found
Applied Advanced Error Control Coding for General Purpose Representation and Association Machine Systems
General-Purpose Representation and Association Machine (GPRAM) is proposed to be focusing on computations in terms of variation and flexibility, rather than precision and speed. GPRAM system has a vague representation and has no predefined tasks. With several important lessons learned from error control coding, neuroscience and human visual system, we investigate several types of error control codes, including Hamming code and Low-Density Parity Check (LDPC) codes, and extend them to different directions. While in error control codes, solely XOR logic gate is used to connect different nodes. Inspired by bio-systems and Turbo codes, we suggest and study non-linear codes with expanded operations, such as codes including AND and OR gates which raises the problem of prior-probabilities mismatching. Prior discussions about critical challenges in designing codes and iterative decoding for non-equiprobable symbols may pave the way for a more comprehensive understanding of bio-signal processing. The limitation of XOR operation in iterative decoding with non-equiprobable symbols is described and can be potentially resolved by applying quasi-XOR operation and intermediate transformation layer. Constructing codes for non-equiprobable symbols with the former approach cannot satisfyingly perform with regarding to error correction capability. Probabilistic messages for sum-product algorithm using XOR, AND, and OR operations with non-equiprobable symbols are further computed. The primary motivation for the constructing codes is to establish the GPRAM system rather than to conduct error control coding per se. The GPRAM system is fundamentally developed by applying various operations with substantial over-complete basis. This system is capable of continuously achieving better and simpler approximations for complex tasks. The approaches of decoding LDPC codes with non-equiprobable binary symbols are discussed due to the aforementioned prior-probabilities mismatching problem. The traditional Tanner graph should be modified because of the distinction of message passing to information bits and to parity check bits from check nodes. In other words, the message passing along two directions are identical in conventional Tanner graph, while the message along the forward direction and backward direction are different in our case. A method of optimizing signal constellation is described, which is able to maximize the channel mutual information. A simple Image Processing Unit (IPU) structure is proposed for GPRAM system, to which images are inputted. The IPU consists of a randomly constructed LDPC code, an iterative decoder, a switch, and scaling and decision device. The quality of input images has been severely deteriorated for the purpose of mimicking visual information variability (VIV) experienced in human visual systems. The IPU is capable of (a) reliably recognizing digits from images of which quality is extremely inadequate; (b) achieving similar hyper-acuity performance comparing to human visual system; and (c) significantly improving the recognition rate with applying randomly constructed LDPC code, which is not specifically optimized for the tasks
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
Power and Bandwidth Efficient Coded Modulation for Linear Gaussian Channels
A scheme for power- and bandwidth-efficient communication on the linear Gaussian channel is proposed. A scenario is assumed in which the channel is stationary in time and the channel characteristics are known at the transmitter. Using interleaving, the linear Gaussian channel with its intersymbol interference is decomposed into a set of memoryless subchannels. Each subchannel is further decomposed into parallel binary memoryless channels, to enable the use of binary codes. Code bits from these parallel binary channels are mapped to higher-order near-Gaussian distributed constellation symbols. At the receiver, the code bits are detected and decoded in a multistage fashion. The scheme is demonstrated on a simple instance of the linear Gaussian channel. Simulations show that the scheme achieves reliable communication at 1.2 dB away from the Shannon capacity using a moderate number of subchannels
Design and Analysis of Nonbinary LDPC Codes for Arbitrary Discrete-Memoryless Channels
We present an analysis, under iterative decoding, of coset LDPC codes over
GF(q), designed for use over arbitrary discrete-memoryless channels
(particularly nonbinary and asymmetric channels). We use a random-coset
analysis to produce an effect that is similar to output-symmetry with binary
channels. We show that the random selection of the nonzero elements of the
GF(q) parity-check matrix induces a permutation-invariance property on the
densities of the decoder messages, which simplifies their analysis and
approximation. We generalize several properties, including symmetry and
stability from the analysis of binary LDPC codes. We show that under a Gaussian
approximation, the entire q-1 dimensional distribution of the vector messages
is described by a single scalar parameter (like the distributions of binary
LDPC messages). We apply this property to develop EXIT charts for our codes. We
use appropriately designed signal constellations to obtain substantial shaping
gains. Simulation results indicate that our codes outperform multilevel codes
at short block lengths. We also present simulation results for the AWGN
channel, including results within 0.56 dB of the unconstrained Shannon limit
(i.e. not restricted to any signal constellation) at a spectral efficiency of 6
bits/s/Hz.Comment: To appear, IEEE Transactions on Information Theory, (submitted
October 2004, revised and accepted for publication, November 2005). The
material in this paper was presented in part at the 41st Allerton Conference
on Communications, Control and Computing, October 2003 and at the 2005 IEEE
International Symposium on Information Theor
Rank Minimization over Finite Fields: Fundamental Limits and Coding-Theoretic Interpretations
This paper establishes information-theoretic limits in estimating a finite
field low-rank matrix given random linear measurements of it. These linear
measurements are obtained by taking inner products of the low-rank matrix with
random sensing matrices. Necessary and sufficient conditions on the number of
measurements required are provided. It is shown that these conditions are sharp
and the minimum-rank decoder is asymptotically optimal. The reliability
function of this decoder is also derived by appealing to de Caen's lower bound
on the probability of a union. The sufficient condition also holds when the
sensing matrices are sparse - a scenario that may be amenable to efficient
decoding. More precisely, it is shown that if the n\times n-sensing matrices
contain, on average, \Omega(nlog n) entries, the number of measurements
required is the same as that when the sensing matrices are dense and contain
entries drawn uniformly at random from the field. Analogies are drawn between
the above results and rank-metric codes in the coding theory literature. In
fact, we are also strongly motivated by understanding when minimum rank
distance decoding of random rank-metric codes succeeds. To this end, we derive
distance properties of equiprobable and sparse rank-metric codes. These
distance properties provide a precise geometric interpretation of the fact that
the sparse ensemble requires as few measurements as the dense one. Finally, we
provide a non-exhaustive procedure to search for the unknown low-rank matrix.Comment: Accepted to the IEEE Transactions on Information Theory; Presented at
IEEE International Symposium on Information Theory (ISIT) 201
Generalized Low-Density Parity-Check Coding Aided Multilevel Codes
Classic Low-Density Parity-Check (LDPC) codes have recently been used as component codes in Multilevel Coding (MLC) due to their impressive BER performance as well as owing to their flexible coding rates. In this paper, we proposed a Multilevel Coding invoking Generalized Low-Density Parity-Check (GLDPC) component codes, which is capable of outperforming the classic LDPC component codes at a reduced decoding latency, when communicating over AWGN and uncorrelated Rayleigh fading channels
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