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Analysis of low-density parity-check codes for the Gilbert-Elliott channel
- IEEE TRANS. INF. THEORY
, 2005
"... Density evolution analysis of low-density parity-check (LDPC) codes in memoryless channels is extended to the Gilbert–Elliott (GE) channel, which is a special case of a large class of channels with hidden Markov memory. In a procedure referred to as estimation decoding, the sum–product algorithm (S ..."
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Cited by 34 (8 self)
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Density evolution analysis of low-density parity-check (LDPC) codes in memoryless channels is extended to the Gilbert–Elliott (GE) channel, which is a special case of a large class of channels with hidden Markov memory. In a procedure referred to as estimation decoding, the sum–product algorithm (SPA) is used to perform LDPC decoding jointly with channel-state detection. Density evolution results show (and simulation results confirm) that such decoders provide a significantly enlarged region of successful decoding within the GE parameter space, compared with decoders that do not exploit the channel memory. By considering a variety of ways in which a GE channel may be degraded, it is shown how knowledge of the decoding behavior at a single point of the GE parameter space may be extended to a larger region within the space, thereby mitigating the large complexity needed in using density evolution to explore the parameter space point-by-point. Using the GE channel as a straightforward example, we conclude that analysis of estimation decoding for LDPC codes is feasible in channels with memory, and that such analysis shows large potential gains.
Density Evolution for Asymmetric Memoryless Channels
- 3rd International Symposium on Turbo Codes and Related Topics
"... Abstract — 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 t ..."
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Abstract — 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. Index Terms — Low-density parity-check (LDPC) codes, density evolution, sum-product algorithm, asymmetric channels, z-channels, rank of random matrices. I.
Joint Iterative Decoding of LDPC Codes and Channels with Memory
, 2003
"... This paper considers the joint iterative decoding of irregular low-density parity-check (LDPC) codes and channels with memory. It begins by introducing a new class of erasure channels with memory, known as generalizederasure channels. For these channels, a single parameter recursion for the density ..."
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Cited by 11 (6 self)
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This paper considers the joint iterative decoding of irregular low-density parity-check (LDPC) codes and channels with memory. It begins by introducing a new class of erasure channels with memory, known as generalizederasure channels. For these channels, a single parameter recursion for the density evolution of the joint iterative decoder is derived. This provides a necessary and sucient condition for decoder convergence, and allows the algebraic construction of sequences of LDPC degree distributions. Under certain conditions, these sequences can achieve the symmetric information rate (SIR) of the channel using only iterative decoding. Example code sequences are given for two channels, and it is conjectured that they each achieve the respective SIR. Keywords: joint iterative decoding, erasure channel, capacityachieving, LDPC codes 1.
Conditionally Cycle-Free Generalized Tanner Graphs: Theory and Application to High-Rate Serially Concatenated Codes
, 2007
"... ... studied by a number of authors under the rubric of generalized parity-check matrices. This work considers the conditioning of binary hidden variables in such models in order to break all cycles and thus derive optimal soft-in soft-out (SISO) decoding algorithms. Conditionally cycle-free generali ..."
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Cited by 2 (0 self)
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... studied by a number of authors under the rubric of generalized parity-check matrices. This work considers the conditioning of binary hidden variables in such models in order to break all cycles and thus derive optimal soft-in soft-out (SISO) decoding algorithms. Conditionally cycle-free generalized Tanner graphs are shown to imply optimal SISO decoding algorithms for the first order Reed-Muller codes and their duals- the extended Hamming codes- which are substantially less complex than conventional bit-level trellis decoding. The study of low-complexity optimal SISO decoding algorithms for the family of extended Hamming codes is practically motivated. Specifically, it is shown that exended Hamming codes offer an attractive alternative to highrate convolutional codes in terms of both performance and complexity for use in very high-rate, very low-floor, serially concatenated coding schemes.
A partial ordering of general finite-state Markov channels under LDPC decoding
- IEEE Trans. Inform. Theory
"... Abstract—A partial ordering on general finite-state Markov channels is given, which orders the channels in terms of probability of symbol error under iterative estimation decoding of a low-density parity-check (LDPC) code. This result is intended to mitigate the complexity of characterizing the perf ..."
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Abstract—A partial ordering on general finite-state Markov channels is given, which orders the channels in terms of probability of symbol error under iterative estimation decoding of a low-density parity-check (LDPC) code. This result is intended to mitigate the complexity of characterizing the performance of general finite-state Markov channels, which is difficult due to the large parameter space of this class of channel. An analysis tool, originally developed for the Gilbert–Elliott channel, is extended and generalized to general finite-state Markov channels. In doing so, an operator is introduced for combining finite-state Markov channels to create channels with larger state alphabets, which are then subject to the partial ordering. As a result, the probability of symbol error performance of finite-state Markov channels with different numbers of states and wide ranges of parameters can be directly compared. Several examples illustrating the use of the techniques are provided, focusing on binary finite-state Markov channels and Gaussian finite-state Markov channels. Furthermore, this result is used to order Gilbert–Elliott channels with different marginal state probabilities, which was left as an open problem by previous work. Index Terms—Estimation-decoding, iterative decoding, lowdensity parity-check (LDPC) codes, Markov channels, partial ordering. I.
Packet-LDPC codes for tape drives
- IEEE Trans. Magn
, 2005
"... In this paper, we introduce packet low-density parity-check (packet-LDPC) codes for high-density tape storage systems. We report on the performance of two error control code (ECC) architectures based on the packet-LDPC codes. The architectures are designed to be (approximately) compatible with the w ..."
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In this paper, we introduce packet low-density parity-check (packet-LDPC) codes for high-density tape storage systems. We report on the performance of two error control code (ECC) architectures based on the packet-LDPC codes. The architectures are designed to be (approximately) compatible with the widely used ECMA-319 ECC standard based on two interleaved concatenated 8-bit Reed–Solomon (RS) codes. One architecture employs an inner RS code; the other employs an inner turbo product code with single parity-check con-stituent codes (TPC-SPC). Both employ a packet-LDPC code as the outer code. As for the ECMA-319 system, both schemes are required to correct noise bursts due to media defects and synchronization loss, as well as the loss of one of eight tracks (due to a head clog, for example). We show that the first packet-LDPC code architecture substantially outperforms the ECMA-319 scheme and is only a few tenths of a decibel inferior to a long, highly complex 12-bit RS scheme. The second architecture outperforms both the ECMA-319 and the long RS code scheme. Index Terms—Error control code (ECC), low-density parity-check (LDPC), packet-LDPC, tape recording. I.
A Class of High-Rate, Low-Complexity, Well-Structured LDPC Codes from Combinatorial Designs and Their Applications on ISI Channels
, 2002
"... We present a systematic construction of a class of highrate, well-structured low density parity check (LDPC) codes based on combinatorial designs. We show that the proposed (2#, # 1})-design results in a class of (2, #)-regular LDPC codes, which are systematic, quasicyclic, free of length-4 and ..."
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Cited by 1 (0 self)
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We present a systematic construction of a class of highrate, well-structured low density parity check (LDPC) codes based on combinatorial designs. We show that the proposed (2#, # 1})-design results in a class of (2, #)-regular LDPC codes, which are systematic, quasicyclic, free of length-4 and length-6 cycles, linear-time encodable and decodable, and which have high code rates of R= (1- # ) . Analysis from the maximum likelihood perspective shows that the distance spectrum of the proposed LDPC codes are better than that of the Gallager ensemble codes for the same code length and rate. The proposed codes are then applied to several inter-symbol interference channels, where 2 high code rates and 3 block sizes from short to mediumn are evaluated. For best performance gain, the i.i.d. capacity is computed to choose the best precoder and iterative decoding and equalization is performed The proposed LDPC codes demonstrate performance that is slightly (but noticeably) better than an average random LDPC code of column weight 3. Unlike random codes, well-structured LDPC codes can lend themselves to a very low-complexity implementation for highspeed applications.
Constant-time algorithm for computing the Euclidean distance maps of binary images on 2D meshes with reconfigurable buses
- Information Science
, 1999
"... Abstract—We present an efficient algorithm to compute the distance spectrum of a general finite intersymbol interference (ISI) channel, whose complexity is lower than those of existing methods. Closed-form expressions are derived for both input–output Euclidean distance enumerators and asymptotic di ..."
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Abstract—We present an efficient algorithm to compute the distance spectrum of a general finite intersymbol interference (ISI) channel, whose complexity is lower than those of existing methods. Closed-form expressions are derived for both input–output Euclidean distance enumerators and asymptotic distance spectrum shapes for 2-tap and 3-tap ISI channels. Coded and/or precoded ISI channels are also discussed. Index Terms—Distance spectrum, input–output Euclidean distance enumerator (IOEDE), input–output weight enumerator, intersymbol interference (ISI) channel, precoding. I.
Channel Matched Iterative Decoding for Magnetic Recording Systems
, 2009
"... iAcknowledgements I will always be in debt to my PhD adviser, Professor Jaekyun Moon, who introduced me to the area of signal processing and coding for the magnetic recording channel. It is only through his immense support, deep intuition, and thorough insightful advise that this work is made possib ..."
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iAcknowledgements I will always be in debt to my PhD adviser, Professor Jaekyun Moon, who introduced me to the area of signal processing and coding for the magnetic recording channel. It is only through his immense support, deep intuition, and thorough insightful advise that this work is made possible. His mark in my academic, professional, and personal lives will never be forgotten. I am also very grateful to Profs Tom Luo, and Nihar Jindal from the Electrical and Computer Engineering department and Prof Paul Garrett from the Mathematics department at the University of Minnesota for serving as committee members in my preliminary and final oral exams. I acknowledge the enriching technical discussions I had with my friends and colleagues in the Communications and Data Storage (CDS) lab, including Jihoon Park, Farshid Rafiee, Jaewook Lee, Seongwook Jeong, and Daejung Yoon. My experience at graduate school was made enjoyable through the good times I spent with my close friends at the university Minnesota, and through the encouragement of old
Conditionally Cycle-Free Generalized Tanner Graphs: Theory and Application to High-Rate Serially Concatenated Codes
, 2006
"... Abstract — Generalized Tanner graphs have been implicitly studied by a number of authors under the rubric of generalized parity-check matrices. This work considers the conditioning of binary hidden variables in such models in order to break all cycles and thus derive optimal soft-in soft-out (SISO) ..."
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Abstract — Generalized Tanner graphs have been implicitly studied by a number of authors under the rubric of generalized parity-check matrices. This work considers the conditioning of binary hidden variables in such models in order to break all cycles and thus derive optimal soft-in soft-out (SISO) decoding algorithms. Conditionally cycle-free generalized Tanner graphs are shown to imply optimal SISO decoding algorithms for the first order Reed-Muller codes and their duals- the extended Hamming codes- which are substantially less complex than conventional bit-level trellis decoding. The study of low-complexity optimal SISO decoding algorithms for the family of extended Hamming codes is practically motivated. Specifically, it is shown that exended Hamming codes offer an attractive alternative to highrate convolutional codes in terms of both performance and complexity for use in very high-rate, very low-floor, serially concatenated coding schemes. I.
