Low-density Generator Matrix Research Articles

Robust Multiresolution Coding

In multiresolution coding a source sequence is encoded into a base layer and a refinement layer. The refinement layer, constructed using a conditional codebook, is in general not decodable without the correct reception of the base layer. By relating multiresolution coding with multiple description coding, we show that it is in fact possible to construct multiresolution codes in certain ways so that the refinement layer alone can be used to reconstruct the source to achieve a nontrivial distortion. As a consequence, one can improve the robustness of the existing multiresolution coding schemes without sacrificing the efficiency. Specifically, we obtain an explicit expression of the minimum distortion achievable by the refinement layer for arbitrary finite alphabet sources with Hamming distortion measure. Experimental results show that the information-theoretic limits can be approached using a practical robust multiresolution coding scheme based on low-density generator matrix codes.

Half-duplex relaying with serially concatenated low-density generator matrix (SCLDGM) codes

Recently, the integration of cooperation with coding has proven to be a very useful technique to enhance relay system performance. Relay systems based on low-density parity-check (LDPC) codes have the potential to approach the theoretical information limits very closely. However, LDPC codes have a disadvantage in that the encoding complexity is high. To solve this problem, serially concatenated low-density generator matrix (SCLDGM) codes are attracting attention owing to their low encoding complexity. However, the performance of SCLDGM coded relaying has not been investigated. In this paper, we apply SCLDGM codes to relay channel with the motivation of lowering the encoding complexity. Since SCLDGM codes are concatenated codes, we cannot use the same method as LDPC coded relaying. We propose a new relaying system and three cooperation protocols suited for SCLDGM codes. Computer simulation results show that the bit error rate (BER) performance of the proposed system is equal to that of conventional LDPC coded relaying system.

Achieving the rate-distortion bound with low-density generator matrix codes

It is shown that binary low-density generator matrix codes can achieve the rate-distortion bound of discrete memoryless sources with general distortion measure via multilevel quantization. A practical encoding scheme based on the survey-propagation algorithm is proposed. The effectiveness of the proposed scheme is verified through simulation.

A fast BER evaluation method for LDGM codes

Abstract Low-density generator matrix (LDGM) codes have recently drawn researchers’ attention thanks to their satisfying performance while requiring only moderate encoding/decoding complexities as well as to their applicability to network codes. In this paper, we aim to propose a fast simulation method useful to investigate the performance of LDGM code. Supported by the confidence interval analysis , the presented method is, for example, 10 8 times quicker than the Monte-Carlo computer simulation for bit-error-rate (BER) in 10 −10 region.

Capacity-Achieving Codes With Bounded Graphical Complexity and Maximum Likelihood Decoding

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> In this paper, the existence of capacity-achieving codes for memoryless binary-input output-symmetric (MBIOS) channels under maximum-likelihood (ML) decoding with bounded graphical complexity is investigated. Graphical complexity of a code is defined as the number of edges in the graphical representation of the code per information bit and is proportional to the decoding complexity per information bit per iteration under iterative decoding. Irregular repeat-accumulate (IRA) codes are studied first. Utilizing the asymptotic average weight distribution (AAWD) of these codes and invoking Divsalar's bound on the binary-input additive white Gaussian noise (BIAWGN) channel, it is shown that simple nonsystematic IRA ensembles outperform systematic IRA and regular low-density parity-check (LDPC) ensembles with the same graphical complexity, and are at most 0.124 dB away from the Shannon limit. However, a conclusive result as to whether these nonsystematic IRA codes can really achieve capacity cannot be reached. Motivated by this inconclusive result, a new family of codes is proposed, called low-density parity-check and generator matrix (LDPC-GM) codes, which are serially concatenated codes with an outer LDPC code and an inner low-density generator matrix (LDGM) code. It is shown that these codes can achieve capacity on any MBIOS channel using ML decoding and also achieve capacity on any BEC using belief propagation (BP) decoding, both with bounded graphical complexity. Moreover, it is shown that, under certain conditions, these capacity-achieving codes have linearly increasing minimum distances and achieve the asymptotic Gilbert–Varshamov bound for all rates. </para>

Lossy Source Compression Using Low-Density Generator Matrix Codes: Analysis and Algorithms

We study the use of low-density generator matrix (LDGM) codes for lossy compression of the Bernoulli symmetric source. First, we establish rigorous upper bounds on the average distortion achieved by check-regular ensemble of LDGM codes under optimal minimum distance source encoding. These bounds establish that the average distortion using such bounded degree families rapidly approaches the Shannon limit as the degrees are increased. Second, we propose a family of message-passing algorithms, ranging from the standard belief propagation algorithm at one extreme to a variant of survey propagation algorithm at the other. When combined with a decimation subroutine and applied to LDGM codes with suitably irregular degree distributions, we show that such a message-passing/decimation algorithm yields distortion very close to the Shannon rate-distortion bound for the binary symmetric source.

MB iterative decoding algorithm on systematic LDGM codes: Performance evaluation

We investigate the performance of regular systematic low-density generator matrix (LDGM) codes under the majority rule based (MB) iterative decoding algorithm. We derive a recursive form which can be used to extract the error performance of the code. Based on the recursive expression, we derive a tight non-recursive lower bound. These results can serve as efficient tools to evaluate the performance of the code for different degrees.

The typical performance of irregular low-density generator-matrix codes for lossy compression

We evaluate the typical performance of irregular low-density generator-matrix (LDGM) codes, which is defined by sparse matrices with arbitrary irregular bit degree distribution and arbitrary check degree distribution, for lossy compression. We apply the replica method under a one-step replica symmetry breaking (1RSB) ansatz to this problem.

Low-Density Graph Codes That Are Optimal for Binning and Coding With Side Information

In this paper, we describe and analyze the source and channel coding properties of a class of sparse graphical codes based on compounding a low-density generator matrix (LDGM) code with a low-density parity-check (LDPC) code. Our first pair of theorems establishes that there exist codes from this ensemble, with all degrees remaining bounded independently of block length, that are simultaneously optimal for both channel coding and source coding with binary data when encoding and decoding are performed optimally. More precisely, in the context of lossy compression, we prove that finite-degree constructions can achieve any pair (R, D) on the rate-distortion curve of the binary symmetric source. In the context of channel coding, we prove that the same finite-degree codes can achieve any pair (C, p) on the capacity-noise curve of the binary symmetric channel (BSC). Next, we show that our compound construction has a nested structure that can be exploited to achieve the Wyner-Ziv bound for source coding with side information (SCSI), as well as the Gelfand-Pinsker bound for channel coding with side information (CCSI). Although the results described here are based on optimal encoding and decoding, the proposed graphical codes have sparse structure and high girth that renders them well suited to message passing and other efficient decoding procedures.

Layered Low-Density Generator Matrix Codes for Super High Definition Scalable Video Coding System

In this paper, we introduce layered low-density generator matrix (Layered-LDGM) codes for super high definition (SHD) scalable video systems. The layered-LDGM codes maintain the correspondence relationship of each layer from the encoder side to the decoder side. This resulting structure supports partial decoding. Furthermore, the proposed layered-LDGM codes create highly efficient forward error correcting (FEC) data by considering the relationship between each scalable component. Therefore, the proposed layered-LDGM codes raise the probability of restoring the important components. Simulations show that the proposed layered-LDGM codes offer better error resiliency than the existing method which creates FEC data for each scalable component independently. The proposed layered-LDGM codes support partial decoding and raise the probability of restoring the base component. These characteristics are very suitable for scalable video coding systems.

Rate-compatible low-density generator matrix codes

We propose a family of rate-compatible codes based on the concatenation of two linear codes with low-density generator matrix, which are a special class of LDPC codes with low encoding complexity. The proposed scheme is characterized by its simplicity of construction, and does not require optimization of the puncturing pattern.

Maximum a Posteriori Estimation With Vector Autoregressive Models for Digital Magnetic Recording Channels

In recent signal processing schemes of various high density digital magnetic storage systems, it needs to detect signal sequences with signal-dependent media noise and colored Gaussian noise, and so on. The more the areal recording density of storage systems gets increasingly, the more it seems increasingly difficult for any signal processing system to reduce or cancel the effects caused by noise and interference because total noise for which several different distributions are mixed occurs frequently in recording channels. High areal density recording needs not only the severe demand for signal detection, but also comes in predisposed to trend for recording by a large-sector size instead of a single sector which consists of 512 information 8-bit bytes. From this trend, nonbinary low-density parity check (LDPC) codes will be important for future recording systems. For these future problems, this paper proposes the signal estimation method based on statistical inference for such a finite mixture model with known number of noise components. Our signal detection scheme with vector (multivariate) autoregressive (AR) models for total noise is applied to maximum a posteriori probability sequence detection. Furthermore, burst error correcting nonbinary low-density generator matrix (LDGM) codes are used for an error correcting code which satisfies the specific run-length limited condition in the proposed signal processing system. We show that the scheme of these error correcting and signal detection methods are effective to estimate signal sequences degraded by a mixture of noise and improve the error rate performances with respect to the conventional scheme using binary LDGM codes and univariate AR models.

On Analysis and Design of Low Density Generator Matrix Codes for Continuous Phase Modulation

We investigate the analysis and design of low density generator matrix (LDGM) codes for continuous phase modulation (CPM). The system uses LDGM codes as an outer code for CPM. For additive white Gaussian noise channels, we derive the union bound to analyze the error floor performance. Design principles for lowering error floors are suggested from this analysis. We propose a design approach of jointly considering the LDGM code degree and the CPM modulation index. Then we consider the rate-adaptive system for slowly fading channels. By changing the rate of the LDGM codes, the information rate of the CPM signals is adapted according to channel variations. We use a low-rate LDGM code as the mother code. Higher rates are achieved by puncturing the output of these codes. To exploit the rate-flexible property of punctured LDGM codes, a rate function is proposed to calculate the rate of each transmitted block. Thus, we can have a quasi-continuous information rate. Numerical results show that this approach can improve the energy efficiency from a discrete-rate adaptation. Using the rate-adaptive approach, up to 11 dB transmitted energy gain can be achieved from the non-adaptive scheme in the low bit-error-rate region (smaller than 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-3</sup> ) for minimum shift keying (MSK).

Serially-Concatenated Low-Density Generator Matrix (SCLDGM) Codes for Transmission Over AWGN and Rayleigh Fading Channels

Low density generator matrix (LDGM) codes are a particular class of low density parity check (LDPC) codes with very low encoding complexity. Single LDGM codes present high error-floors, which can be substantially reduced with the serial concatenation of two LDGM (SCLDGM) codes. We propose a technique to obtain good SCLDGM codes using extrinsic information transfer (EXIT) functions in a novel way. Although the optimization is performed for AWGN channels with binary signaling, the resulting codes are also optimal for AWGN and perfectly-interleaved Rayleigh fading channels with non-binary signaling and perfect CSI at reception, provided that Gray mapping is utilized. Optimized regular and irregular SCLDGM codes outperform heuristically-designed LDGM codes existing in the literature, and have a performance similar to or better than that of irregular repeat accumulate (IRA) codes.

Serially-Concatenated LDGM Codes for MIMO Channels

We investigate the ability of serially-concatenated low density generator matrix (SCLDGM) codes to approach the capacity of ergodic multiple-input multiple-output (MIMO) channels. First, we explain how to design good SCLDGM codes using extrinsic information transfer (EXIT) analysis and optimum detection. However, optimum detection becomes unfeasible in many cases of practical interest where the number of transmitting antennas and/or the number of bits per constellation symbol is large. In this case, we resort to list sphere detection (LSD), considering two versions: maximum likelihood (ML) and maximum a posteriori (MAP). Again, we use EXIT analysis to obtain good SCLDGM codes for these specific LSD versions. We provide examples of optimized rate 1/2 SCLDGM codes and show that they perform within 1 dB from the theoretical capacity limit for a large number of antenna configurations and modulation formats. Finally, we discuss how to iteratively improve channel estimation using the soft information computed during the decoding process. We also show how the complexity of channel estimation is reduced by utilizing the candidate list calculated by the LSD.

Low-density generator matrix codes for indoor and markov channels

We propose a modified algorithm for decoding of linear codes with low-density generator matrix (LDGM codes) over finite-state binary Markov channels. In order to avoid error floors, a serial concatenation of two LDGM codes is utilized. The hidden Markov model representing the channel is incorporated into the graph corresponding to the code, and the message passing algorithm is modified accordingly. The proposed scheme clearly outperforms systems in which the channel statistics are not exploited in the decoding process, allowing reliable communication at rates which are above the capacity of a memoryless channel with the same stationary bit error probability as the Markov channel. The proposed technique can be successfully applied for real wireless channels that can be modeled with hidden Markov models, such as indoor channels. In this case, the hidden Markov model representing the wireless channel can be estimated jointly with the decoding process

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In this letter, we investigate the concatenation of a low density generator matrix (LDGM) outer code and a recursive space time trellis (RSTTC) inner code based on differential encoding. We provide guidelines for the design of the LDGM outer code and observe that previous schemes based on parity check outer codes are particular cases of the proposed framework

Tight Bounds for LDPC and LDGM Codes Under MAP Decoding

A new method for analyzing low density parity check (LDPC) codes and low density generator matrix (LDGM) codes under bit maximum a posteriori probability (MAP) decoding is introduced. The method is based on a rigorous approach to spin glasses developed by Francesco Guerra. It allows to construct lower bounds on the entropy of the transmitted message conditional to the received one. Based on heuristic statistical mechanics calculations, we conjecture such bounds to be tight. The result holds for standard irregular ensembles when used over binary input output symmetric channels. The method is first developed for Tanner graph ensembles with Poisson left degree distribution. It is then generalized to `multi-Poisson' graphs, and, by a completion procedure, to arbitrary degree distribution.

LDGM Codes for Channel Coding and Joint Source-Channel Coding of Correlated Sources

We propose a coding scheme based on the use of systematic linear codes with low-density generator matrix (LDGM codes) for channel coding and joint source-channel coding of multiterminal correlated binary sources. In both cases, the structures of the LDGM encoder and decoder are shown, and a concatenated scheme aimed at reducing the error floor is proposed. Several decoding possibilities are investigated, compared, and evaluated. For different types of noisy channels and correlation models, the resulting performance is very close to the theoretical limits.

Correlated Sources Over Wireless Channels: Cooperative Source-Channel Coding

We consider wireless sensor networks deployed to observe arbitrary random fields. The requirement is to reconstruct an estimate of the random field at a certain collector node. This creates a many-to-one data gathering wireless channel. One of the main challenges in this scenario is that the source/channel separation theorem, proved by Shannon for point-to-point links, does not hold any more. In this paper, we construct novel cooperative source-channel coding schemes that exploit the wireless channel and the correlation between the sources. In particular, we differentiate between two distinct cases. The first case assumes that the sensor nodes are equipped with receivers and, hence, every node can exploit the wireless link to distribute its information to its neighbors. We then devise an efficient deterministic cooperation strategy where the neighboring nodes act as virtual antennas in a beamforming configuration. The second, and more challenging, scenario restricts the capability of sensor nodes to transmit only. In this case, we argue that statistical cooperative source-channel coding techniques still yield significant performance gains in certain relevant scenarios. Specifically, we propose a low complexity cooperative source-channel coding scheme based on the proper use of low-density generator matrix codes. This scheme is shown to outperform the recently proposed joint source-channel coding scheme (Garcia-Frias et al., 2002) in the case of highly correlated sources. In both the deterministic and statistical cooperation scenarios, we develop analytical results that guide the optimization of the proposed schemes and validate the performance gains observed in simulations.