Analysis of sum-product decoding of low-density parity-check codes using a Gaussian approximation

Density evolution is an algorithm for computing the capacity of low-density parity-check (LDPC) codes under message-passing decoding. For memoryless binary-input continuous-output additive white Gaussian noise (AWGN) channels and sum-product decoders, we use a Gaussian approximation for message dens...

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Bibliographic Details
Published in:	IEEE transactions on information theory Vol. 47; no. 2; pp. 657 - 670
Main Authors:	Sae-Young Chung, Richardson, T.J., Urbanke, R.L.
Format:	Journal Article
Language:	English
Published:	New York IEEE 01.02.2001 The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:	Algorithms Approximation Binary system Codes Decoding Density Error detection coding Evolution Gaussian Graphs Mathematical analysis Thresholds
ISSN:	0018-9448, 1557-9654
Online Access:	Get full text
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Abstract	Density evolution is an algorithm for computing the capacity of low-density parity-check (LDPC) codes under message-passing decoding. For memoryless binary-input continuous-output additive white Gaussian noise (AWGN) channels and sum-product decoders, we use a Gaussian approximation for message densities under density evolution to simplify the analysis of the decoding algorithm. We convert the infinite-dimensional problem of iteratively calculating message densities, which is needed to find the exact threshold, to a one-dimensional problem of updating the means of the Gaussian densities. This simplification not only allows us to calculate the threshold quickly and to understand the behavior of the decoder better, but also makes it easier to design good irregular LDPC codes for AWGN channels. For various regular LDPC codes we have examined, thresholds can be estimated within 0.1 dB of the exact value. For rates between 0.5 and 0.9, codes designed using the Gaussian approximation perform within 0.02 dB of the best performing codes found so far by using density evolution when the maximum variable degree is 10. We show that by using the Gaussian approximation, we can visualize the sum-product decoding algorithm. We also show that the optimization of degree distributions can be understood and done graphically using the visualization.
AbstractList	Density evolution is an algorithm for computing the capacity of low-density parity-check (LDPC) codes under message-passing decoding. For memoryless binary-input continuous-output additive white Gaussian noise (AWGN) channels and sum-product decoders, we use a Gaussian approximation for message densities under density evolution to simplify the analysis of the decoding algorithm. We convert the infinite-dimensional problem of iteratively calculating message densities, which is needed to find the exact threshold, to a one-dimensional problem of updating the means of the Gaussian densities. This simplification not only allows us to calculate the threshold quickly and to understand the behavior of the decoder better, but also makes it easier to design good irregular LDPC codes for AWGN channels. For various regular LDPC codes we have examined, thresholds can be estimated within 0.1 dB of the exact value. For rates between 0.5 and 0.9, codes designed using the Gaussian approximation perform within 0.02 dB of the best performing codes found so far by using density evolution when the maximum variable degree is 10. We show that by using the Gaussian approximation, we can visualize the sum-product decoding algorithm. We also show that the optimization of degree distributions can be understood and done graphically using the visualization Density evolution is an algorithm for computing the capacity of low-density parity-check (LDPC) codes under message-passing decoding. For memoryless binary-input continuous-output additive white Gaussian noise (AWGN) channels and sum-product decoders, we use a Gaussian approximation for message densities under density evolution to simplify the analysis of the decoding algorithm. We convert the infinite-dimensional problem of iteratively calculating message densities, which is needed to find the exact threshold, to a one-dimensional problem of updating the means of the Gaussian densities. This simplification not only allows us to calculate the threshold quickly and to understand the behavior of the decoder better, but also makes it easier to design good irregular LDPC codes for AWGN channels. For various regular LDPC codes we have examined, thresholds can be estimated within 0.1 dB of the exact value. For rates between 0.5 and 0.9, codes designed using the Gaussian approximation perform within 0.02 dB of the best performing codes found so far by using density evolution when the maximum variable degree is 10. We show that by using the Gaussian approximation, we can visualize the sum-product decoding algorithm. We also show that the optimization of degree distributions can be understood and done graphically using the visualization. Density evolution is an algorithm for computing the capacity of low-density parity-check (LDPC) codes under message-passing decoding. For memory less binary-input continuous-output additive white Gaussian noise (AWGN) channels and sum-product decoders, we use a Gaussian approximation for message densities under density evolution to simplify the analysis of the decoding algorithm.
Author	Urbanke, R.L. Richardson, T.J. Sae-Young Chung
Author_xml	– sequence: 1 surname: Sae-Young Chung fullname: Sae-Young Chung organization: Lab. for Inf. & Decision Syst., MIT, Cambridge, MA, USA – sequence: 2 givenname: T.J. surname: Richardson fullname: Richardson, T.J. – sequence: 3 givenname: R.L. surname: Urbanke fullname: Urbanke, R.L.
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References	ref13 wozencraft (ref27) 1965 ref12 ref15 ref14 wiberg (ref26) 1996 ref31 frey (ref19) 1999; 12 ref11 rusmevichientong (ref20) 1999; 12 ref1 ref17 ref16 jin (ref30) 2000 divsalar (ref18) 2000 gradshtein (ref28) 1965 gallager (ref2) 1963 chung (ref10) 2000 bertsimas (ref29) 1997 ref24 ref23 ref25 ref22 ref21 ref8 ref7 ref4 chung (ref9) 0 ref3 ref6 ref5
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Snippet	Density evolution is an algorithm for computing the capacity of low-density parity-check (LDPC) codes under message-passing decoding. For memoryless... Density evolution is an algorithm for computing the capacity of low-density parity-check (LDPC) codes under message-passing decoding. For memory less...
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SubjectTerms	Algorithms Approximation Binary system Codes Decoding Density Error detection coding Evolution Gaussian Graphs Mathematical analysis Thresholds
Title	Analysis of sum-product decoding of low-density parity-check codes using a Gaussian approximation
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