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68
On the application of LDPC codes to arbitrary discrete memoryless channels
"... Abstract—We discuss three structures of modified lowdensity paritycheck (LDPC) code ensembles designed for transmission over arbitrary discrete memoryless channels. The first structure is based on the wellknown binary LDPC codes following constructions proposed by Gallager and McEliece, the seco ..."
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Cited by 56 (2 self)
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Abstract—We discuss three structures of modified lowdensity paritycheck (LDPC) code ensembles designed for transmission over arbitrary discrete memoryless channels. The first structure is based on the wellknown binary LDPC codes following constructions proposed by Gallager and McEliece, the second is based on LDPC codes of arbitrary (ary) alphabets employing moduloaddition, as presented by Gallager, and the third is based on LDPC codes defined over the field GF (). All structures are obtained by applying a quantization mapping on a coset LDPC ensemble. We present tools for the analysis of nonbinary codes and show that all configurations, under maximumlikelihood (ML) decoding, are capable of reliable communication at rates arbitrarily close to the capacity of any discrete memoryless channel. We discuss practical iterative decoding of our structures and present simulation results for the additive white Gaussian noise (AWGN) channel confirming the effectiveness of the codes. Index Terms —ary lowdensity parity check (LDPC), belief propagation, coset codes, iterative decoding, LDPC codes, turbo codes. I.
Capacityapproaching bandwidthefficient coded modulation schemes based on lowdensity paritycheck codes
 IEEE Trans. on Information Theory
, 2003
"... Abstract—We design multilevel coding (MLC) and bitinterleaved coded modulation (BICM) schemes based on lowdensity paritycheck (LDPC) codes. The analysis and optimization of the LDPC component codes for the MLC and BICM schemes are complicated because, in general, the equivalent binaryinput compo ..."
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Cited by 53 (5 self)
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Abstract—We design multilevel coding (MLC) and bitinterleaved coded modulation (BICM) schemes based on lowdensity paritycheck (LDPC) codes. The analysis and optimization of the LDPC component codes for the MLC and BICM schemes are complicated because, in general, the equivalent binaryinput component channels are not necessarily symmetric. To overcome this obstacle, we deploy two different approaches: one based on independent and identically distributed (i.i.d.) channel adapters and the other based on coset codes. By incorporating i.i.d. channel adapters, we can force the symmetry of each binaryinput component channel. By considering coset codes, we extend the concentration theorem based on previous work by Richardson et al. and Kavčić et al. We also discuss the relation between the systems based on the two approaches and show that they indeed have the same expected decoder behavior. Next, we jointly optimize the code rates and degree distribution pairs of the LDPC component codes for the MLC scheme. The optimized irregular LDPC codes at each level of MLC with multistage decoding (MSD) are able to perform well at signaltonoise ratios (SNR) very close to the capacity of the additive white Gaussian noise (AWGN) channel. We also show that the optimized BICM scheme can approach the parallel independent decoding (PID) capacity as closely as does the MLC/PID scheme. Simulations with very large codeword length verify the accuracy of the analytical results. Finally, we compare the simulated performance of these coded modulation schemes at finite codeword lengths, and consider the results from the perspective of a random coding exponent analysis. Index Terms—Bitinterleaved coded modulation (BICM), coding exponent analysis, coset codes, density evolution, independent and identically distributed (i.i.d.) channel adapters, irregular lowdensity paritycheck (LDPC) codes, LDPC codes, multilevel coding (MLC). I.
Design and analysis of nonbinary LDPC codes for arbitrary discretememoryless channels
 IEEE TRANS. INFORM. THEORY
, 2005
"... We present an analysis, under iterative decoding, of coset LDPC codes over GF(q), designed for use over arbitrary discretememoryless channels (particularly nonbinary and asymmetric channels). We use a randomcoset analysis to produce an effect that is similar to outputsymmetry with binary channels ..."
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Cited by 44 (1 self)
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We present an analysis, under iterative decoding, of coset LDPC codes over GF(q), designed for use over arbitrary discretememoryless channels (particularly nonbinary and asymmetric channels). We use a randomcoset analysis to produce an effect that is similar to outputsymmetry with binary channels. We show that the random selection of the nonzero elements of the GF(q) paritycheck matrix induces a permutationinvariance 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.
Analysis of lowdensity paritycheck codes for the GilbertElliott channel
 IEEE TRANS. INF. THEORY
, 2005
"... Density evolution analysis of lowdensity paritycheck (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 lowdensity paritycheck (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 channelstate 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 pointbypoint. 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 lowdensity paritycheck (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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Cited by 31 (5 self)
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Abstract — Density evolution is one of the most powerful analytical tools for lowdensity paritycheck (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 nonsymmetric memoryless channels, which in turn broadens the applications to general memoryless channels, e.g. zchannels, 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 — Lowdensity paritycheck (LDPC) codes, density evolution, sumproduct algorithm, asymmetric channels, zchannels, rank of random matrices. I.
On the application of factor graphs and the sumproduct algorithm to ISI channels
 IEEE Trans. Commun
, 2005
"... Abstract—In this paper, based on the application of the sum–product (SP) algorithm to factor graphs (FGs) representing the joint a posteriori probability (APP) of the transmitted symbols, we propose new iterative softinput softoutput (SISO) detection schemes for intersymbol interference (ISI) chan ..."
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Cited by 31 (5 self)
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Abstract—In this paper, based on the application of the sum–product (SP) algorithm to factor graphs (FGs) representing the joint a posteriori probability (APP) of the transmitted symbols, we propose new iterative softinput softoutput (SISO) detection schemes for intersymbol interference (ISI) channels. We have verified by computer simulations that the SP algorithm converges to a good approximation of the exact marginal APPs of the transmitted symbols if the FG has girth at least 6. For ISI channels whose corresponding FG has girth 4, the application of a stretching technique allows us to obtain an equivalent girth6 graph. For sparse ISI channels, the proposed algorithms have advantages in terms of complexity over optimal detection schemes based on the Bahl–Cocke–Jelinek–Raviv (BCJR) algorithm. They also allow a parallel implementation of the receiver and the possibility of a more efficient complexity reduction. The application to joint detection and decoding of lowdensity paritycheck (LDPC) codes is also considered and results are shown for some partialresponse magnetic channels. Also in these cases, we show that the proposed algorithms have a limited performance loss with respect to that can be obtained when the optimal “serial ” BCJR algorithm is used for detection. Therefore, for their parallel implementation, they represent a favorable alternative to the modified “parallel ” BCJR algorithm proposed in the literature for the application to magnetic channels. Index Terms—Factor graphs, intersymbol interference (ISI) channels, iterative detection, lowdensity paritycheck (LDPC) codes, partialresponse channels, sum–product (SP) algorithm. I.
On the Capacity Loss due to Separation of Detection and Decoding
, 2002
"... The performance loss due to separation of detection and decoding on the binaryinput additive white Gaussian noise channel is quantified in terms of mutual information. Results are reported for both the codedivision multipleaccess (CDMA) channel in the large system limit and the intersymbol interf ..."
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Cited by 24 (10 self)
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The performance loss due to separation of detection and decoding on the binaryinput additive white Gaussian noise channel is quantified in terms of mutual information. Results are reported for both the codedivision multipleaccess (CDMA) channel in the large system limit and the intersymbol interference (ISI) channel. The results for CDMA rely on the replica method developed in statistical mechanics. It is shown that a previous result in [1] found for Gaussian input alphabet holds also for binary input alphabets. For the ISI channel, the performance loss is calculated via the BCJR algorithm. Comparisons are made to the capacity of separate detection and decoding using suboptimum detectors such as a decisionfeedback equalizer.
Iterative Detection and Decoding for Separable TwoDimensional Intersymbol Interference
 IEEE Trans. Magnetics
, 2003
"... Abstract—We introduce two detection methods for uncoded twodimensional (2D) intersymbol interference (ISI) channels. The detection methods are suitable for a special case of 2D ISI channels where the channel response is separable. In this case, the 2D ISI is treated as the concatenation of two o ..."
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Cited by 23 (4 self)
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Abstract—We introduce two detection methods for uncoded twodimensional (2D) intersymbol interference (ISI) channels. The detection methods are suitable for a special case of 2D ISI channels where the channel response is separable. In this case, the 2D ISI is treated as the concatenation of two onedimensional ISI channels. The first method uses equalization to reduce the ISI in one of the two dimensions followed by use of a maximum a posteriori (MAP) detector for the ISI in the other dimension. The second method employs modified MAP algorithms to reduce the ISI in each dimension. The implementation complexity of the two methods grows exponentially in the ISI length in either the row or column dimension. We develop two iterative decoding schemes based on these detection methods and lowdensity paritycheck codes as error correction codes. Simulation results show that the biterrorrate performance loss caused by the 2D ISI for the separable channel response considered is less than 1 dB over a channel without ISI. This motivates equalizing a general 2D ISI channel response to a nearby separable matrix. Index Terms—Equalization, iterative decoding, nonbinary MAP, turbo equalization, twodimensional intersymbol interference. I.
On LDPC codes over channels with memory
 IEEE Trans. Wireless Commun
, 2006
"... Abstract — The problem of detection and decoding of lowdensity paritycheck (LDPC) codes transmitted over channels with memory is addressed. A new general method to build a factor graph which takes into account both the code constraints and the channel behavior is proposed and the a posteriori proba ..."
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Cited by 18 (12 self)
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Abstract — The problem of detection and decoding of lowdensity paritycheck (LDPC) codes transmitted over channels with memory is addressed. A new general method to build a factor graph which takes into account both the code constraints and the channel behavior is proposed and the a posteriori probabilities of the information symbols, necessary to implement maximum a posteriori (MAP) symbol detection, are derived by using the sumproduct algorithm. With respect to the case of a LDPC code transmitted on a memoryless channel, the derived factor graphs have additional factor nodes taking into account the channel behavior and not the code constraints. It is shown that the function associated to the generic factor node modeling the channel is related to the basic branch metric used in the Viterbi algorithm when MAP sequence detection is applied or in the BCJR algorithm implementing MAP symbol detection. This fact suggests that all the previously proposed solutions for those algorithms can be systematically extended to LDPC codes and graphbased detection. When the sumproduct algorithm works on the derived factor graphs, the most demanding computation is in general that performed at factor nodes modeling the channel. In fact, the complexity of the computation at these factor nodes is in general exponential in a suitably defined channel memory parameter. In these cases, a technique for complexity reduction is illustrated. In some particular cases of practical relevance, the above mentioned complexity becomes linear in the channel memory. This does not happen in the same cases when detection is performed by using the Viterbi algorithm or the BCJR algorithm, suggesting that the use of factor graphs and the sumproduct algorithm might be computationally more appealing. As an example of application of the described framework, the cases of noncoherent and flat fading channels are considered. Index Terms — Factor graphs, sumproduct algorithm, channels with memory, phasenoise, flat fading, lowdensity paritycheck codes, iterative detection/decoding. I.
Iterative decoding and equalization for 2D recording channels
 IEEE Trans. Magn
, 2002
"... Abstract—We study iterative decoding and equalization for information storage systems that have twodimensional (2D) intersymbol interference (ISI) during readback. Two iterative schemes for 2D equalization are introduced and evaluated. The first is based on minimum mean squared error (MMSE) esti ..."
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Cited by 17 (6 self)
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Abstract—We study iterative decoding and equalization for information storage systems that have twodimensional (2D) intersymbol interference (ISI) during readback. Two iterative schemes for 2D equalization are introduced and evaluated. The first is based on minimum mean squared error (MMSE) estimation and the second is based on message passing on the combined graph of the ISI and the error correction code. Lowdensity paritycheck codes are used for error correction. For the form of the ISI considered in our simulations the best performance is achieved by using the iterative decoding and MMSE equalization method. Index Terms—Equalization, intersymbol interference, iterative decoding, message passing. I.