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Performance analysis of noisy message-passing decoding of low-density parity-check codes

Alla Tarighati, Hamed Farhadi, Farshad Lahouti

Published Jan 1, 2010• 6 citations
Publisher

Abstract

A noisy message-passing decoding scheme isconsidered for low-density parity-check (LDPC) codes overadditive white Gaussian noise (AWGN) channels. The internaldecoder noise is motivated by the quantization noise in digitalimplementations or the intrinsic noise of analog LDPC decoders.We modelled the decoder noise as AWGN on the exchangedmessages in the iterative LDPC decoder. This is shown to renderthe message densities in the noisy LDPC decoder inconsistent.We then invoke Gaussian approximation and formulate a twodimensionaldensity evolution analysis for the noisy LDPCdecoder. This allows for not only tracking the mean, but also thevariance of the message densities, and hence, quantifying thethreshold of the LDPC code. According to the results, a decodernoise of unit variance, increases the threshold for a regular (3,6) code by 1.672dB.

DOI
10.13140/2.1.2792.3527

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A noisy message-passing decoding scheme isconsidered for low-density parity-check (LDPC) codes overadditive white Gaussian noise (AWGN) channels. The internaldecoder noise is motivated by the quantization noise in digitalimplementations or the intrinsic noise of analog LDPC decoders.We modelled the decoder noise as AWGN on the exchangedmessages in the iterative LDPC decoder. This is shown to renderthe message densities in the noisy LDPC decoder inconsistent.We then invoke Gaussian approximation and formulate a twodimensionaldensity evolution analysis for the noisy LDPCdecoder. This allows for not only tracking the mean, but also thevariance of the message densities, and hence, quantifying thethreshold of the LDPC code. According to the results, a decodernoise of unit variance, increases the threshold for a regular (3,6) code by 1.672dB.
We address noisy message-passing decoding of lowdensity parity-check (LDPC) codes over additive white Gaussian noise channels. Message-passing decoders in which certain processing units iteratively exchange messages are common for decoding LDPC codes. The exchanged messages are in general subject to internal noise in hardware implementation of these decoders. We model the internal decoder noise as additive white Gaussian noise (AWGN) degrading exchanged messages. Using Gaussian approximation of the exchanged messages, we perform a two-dimensional density evolution analysis for the noisy LDPC decoder. This makes it possible to track both the mean, and the variance of the exchanged message densities, and hence, to quantify the threshold of the LDPC code in the presence of internal decoder noise. The numerical and simulation results are presented that quantify the performance loss due to the internal decoder noise. To partially compensate this performance loss, we propose a simple method, based on EXIT chart analysis, to design robust irregular LDPC codes. The simulation results indicate that the designed codes can indeed compensate part of the performance loss due to the internal decoder noise.
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.
This paper investigates about the usefulness of some recently published low complexity upper bounds on belief-propagation decoding thresholds for irregular low-density parity-check (LDPC) codes to analyze their performance on memoryless binary input - additive white Gaussian noise (BI-AWGN) channels, with sum-product decoding. Irregular LDPC codes are known to perform better than regular ones, and to exhibit, like them, the so called “threshold phenomenon”, being the threshold defined as the maximum noise level such that an arbitrarily small bit-error probability can be achieved as the block length tends to infinity. We use a simplified analysis of the belief-propagation decoding algorithm, i.e., consider a Gaussian approximation for message densities under density evolution, and a simple algorithmic method, defined recently, to estimate the decoding thresholds for regular and irregular LDPC codes on BI-AWGN channels.
For decoding low-density parity-check (LDPC) codes on discrete memoryless channels, a method to quantize messages and to find message-passing decoding functions for the variable and check nodes is developed. These are used to obtain noise thresholds by density evolution. The message-passing decoding alphabet is restricted to be discrete with a fixed maximum alphabet size. Discrete quantization is required to obtain this fixed alphabet size; a greedy algorithm which uses the mutual information between the code bit and message is presented. It is argued that using this message-passing decoding framework is more efficient for approaching channel capacity than simply quantizing the belief-propagation algorithm. This method is evaluated using regular LDPC codes on the binary symmetric channel. Using a maximum alphabet size of 16 (4 bits), noise thresholds close to those of belief propagation are obtained.
We present the formulas for searching for the thresholds of regular and irregular low-density parity-check (LDPC) codes under message-passing (MP) algorithm. A Gaussian approximation is applied to studying the evolution of the means of the messages of the variable nodes and the check nodes. Accurate numerical integration methods by using transformations are shown for evaluating the expected values of the message of the check nodes. Tables are built first and interpolations are used for further evaluations. Two curves are used to locate the threshold. We utilize an iterative decoding tunnel between these two curves and study the decoding performance by evaluating conditions of the derivatives of these two curves. Using this method the performance of both regular and irregular LDPC codes can be studied in a unified manner without using simulation.
Departing from traditional communication theory where decoding algorithms are assumed to perform without error, a system where noise perturbs both computational devices and communication channels is considered here. This paper studies limits in processing noisy signals with noisy circuits by investigating the effect of noise on standard iterative decoders for low-density parity-check (LDPC) codes. Concentration of decoding performance around its average is shown to hold when noise is introduced into message-passing and local computation. Density evolution equations for simple faulty iterative decoders are derived. In one model, computing nonlinear estimation thresholds shows that performance degrades smoothly as decoder noise increases, but arbitrarily small probability of error is not achievable. Probability of error may be driven to zero in another system model; the decoding threshold again decreases smoothly with decoder noise. As an application of the methods developed, an achievability result for reliable memory systems constructed from unreliable components is provided.