Variations on the Gallager bounds, connections, and applications

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, zchannels, rank of random matrices.

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“…[24], [25], and [26]). Various Gallager type bounds on ML decoders for different finite LDPC code ensembles have been established in [27].…”

mentioning

confidence: 99%

Density Evolution for Asymmetric Memoryless Channels

Wang

Kulkarni²,

Poor³

2005

IEEE Trans. Inform. Theory

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“…In [21], a simpler proof of the SFB is derived, and by doing so, the simplified proof reproduces the SFB as a particular case of the DS2 bound (see Eq. (4)).…”

Section: The Shulman and Feder Boundmentioning

confidence: 99%

“…This simplification was also done in [21] for the derivation of the SFB as a particular case of the DS2 bound. When considering the case of an upper bound on the block error probability, this simplification is reasonable because we consider the terms…”

Section: Upper Bounds On Bit Error Probabilitymentioning

confidence: 99%

“…Here, ψ m N (y) is an arbitrary probability tilting measure (which may depend on the transmitted codeword c m ), and p N (y|c) designates the transition probability measure of the channel. Connections between these two seemingly different bounding techniques in (1) and (2) were demonstrated in [21], showing that many previously reported bounds (or their Chernoff versions) whose derivation originally relied on the concept shown in inequality (1) can in fact be re-produced as particular cases of the bounding technique used in (2). To this end, one simply needs to choose the suitable probability tilting measure ψ m N which serves as the "kernel" for reproducing various previously reported bounds.…”

Section: Introductionmentioning

confidence: 99%

“…However, in the random coding setting, it fails to reproduce the random coding exponent [10] while the second version of the Duman and Salehi (DS2) bound, to be reviewed in the next section, does (see [21]). The Shulman-Feder bound (SFB) can be derived as a particular case of the DS2 bound (see [21]), and it achieves the random coding error exponent.…”

Section: Introductionmentioning

confidence: 99%

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Tightened Upper Bounds on the ML Decoding Error Probability of Binary Linear Block Codes

Twitto

Sason

Shamai

2006

2006 IEEE International Symposium on Information Theory

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The performance of maximum-likelihood (ML) decoded binary linear block codes is addressed via the derivation of tightened upper bounds on their decoding error probability. The upper bounds on the block and bit error probabilities are valid for any memoryless, binary-input and outputsymmetric communication channel, and their effectiveness is exemplified for various ensembles of turbo-like codes over the AWGN channel. An expurgation of the distance spectrum of binary linear block codes further tightens the resulting upper bounds.

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Gallager bounds on the performance of maximum‐likelihood decoded linear binary block codes in AWGN interference

Yousefi

2006

Trans Emerging Tel Tech

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SUMMARYPerformance evaluation of maximum-likelihood (ML) soft-decision-decoded binary block codes is usually carried out using bounding techniques. Many tight upper bounds on the error probability of binary codes are based on the so-called Gallager's first bounding technique (GFBT). In this method, Gallager bisects the error probability to the joint probability of error and noise residing in a region < (here referred to as the Gallager region) plus joint probability of error and noise residing in the complement of < (also referred to as regions of many and few errors, respectively); where < is a volume around the transmitted codeword. In this tutorial review, a comprehensive study of a number of upper bounds on the error probability of ML decoding of binary codes based on GFBT is provided. For some bounds, their applicability to other schemes is also pointed out and argued. We also provide an overview of bounds based on Gallager's second bounding technique (GSBT) and discuss some of their relations and interconnections.

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Variations on the Gallager bounds, connections, and applications

Cited by 127 publications

References 27 publications

Density Evolution for Asymmetric Memoryless Channels

Density Evolution for Asymmetric Memoryless Channels

Tightened Upper Bounds on the ML Decoding Error Probability of Binary Linear Block Codes

Gallager bounds on the performance of maximum‐likelihood decoded linear binary block codes in AWGN interference

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