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SiPS ... design and implementation - IEEE Workshop on Signal Processing Systems

Optimal Output Quantization of Binary Input AWGN Channel for Belief-Propagation Decoding of LDPC Codes

Junho Cho, Jonghong Kim, Wonyong Sung

Published Oct 1, 2012• 5 citations
Publisher

Abstract

Optimal quantizers for channel output are especially necessary for very high speed and memory-cell sensing applications where high-precision analog-to-digital converters are not available. This paper investigates a method to find optimal output quantizers for a binary input (BI) additive white Gaussian noise (AWGN) channel when regular low-density parity-check (LDPC) codes are used with belief-propagation (BP) decoding. The optimal quantizers are found by uni-parametric and multi-parametric optimization methods based on the discretized density evolution (DE). The capacity of various LDPC code ensembles are also provided with the BI-AWGN channel outputs quantized using these optimal quantizers. It is demonstrated that only 3 bits of quantization precision performs almost within 0.1 dB of the infinite precision channel output. The effect of the code rate and the degree distribution of LDPC codes on the optimal quantization boundaries is analyzed.

DOI
10.1109/sips.2012.51

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Optimal quantizers for channel output are especially necessary for very high speed and memory-cell sensing applications where high-precision analog-to-digital converters are not available. This paper investigates a method to find optimal output quantizers for a binary input (BI) additive white Gaussian noise (AWGN) channel when regular low-density parity-check (LDPC) codes are used with belief-propagation (BP) decoding. The optimal quantizers are found by uni-parametric and multi-parametric optimization methods based on the discretized density evolution (DE). The capacity of various LDPC code ensembles are also provided with the BI-AWGN channel outputs quantized using these optimal quantizers. It is demonstrated that only 3 bits of quantization precision performs almost within 0.1 dB of the infinite precision channel output. The effect of the code rate and the degree distribution of LDPC codes on the optimal quantization boundaries is analyzed.
The performance of low-density parity-check (LDPC) codes transmitted over a memoryless binary-input continuous output additive white Gaussian noise (AWGN) channel and decoded with quantized min-sum decoding is strongly influenced by the decoder's quantization scheme. This paper presents an efficient algorithm that determines the best uniform scalar quantizer for a particular code. To maximize performance, it is necessary to determine degree distributions that best match the characteristics of the quantized min-sum decoder. Toward this end, an iterative optimization framework that jointly optimizes the degree distributions and the quantizer is presented
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Designing realizable Maximum Likelihood (ML) decoders for low-density parity-check (LDPC) codes has always been a challenging problem. Maxwell decoder is a well-known ML decoder for LDPC codes over a binary erasure channel (BEC). Based on the concept used in the Maxwell decoder, it may be possible to design near-ML decoders for LDPC codes over an additive white Gaussian noise (AWGN) channel. In this paper, we introduce a novel idea called channel transformation, which will be used as a bridge linking a BEC channel and an AWGN channel. With the transformation, we further propose a near-ML decoder for LDPC codes sent over an AWGN channel. The proposed near-ML decoder, combining a Belief Propagation (BP) decoder and a Maxwell decoder, is called the BP-Maxwell (BM) decoder. Simulation results show that compared with a BP decoder, the BM decoding algorithm can accomplish better frame error rates (FERs) by eliminating many of the error frames with a smaller percentage of error bits. Thus, the BM decoder is a step closer to the ML decoder compared with the BP decoder.
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 extends the work on density evolution for binary low-density parity-check (LDPC) codes with Gaussian approximation to LDPC codes over GF(q) . We first generalize the definition of channel symmetry for nonbinary inputs to include q-ary phase-shift keying (PSK) modulated channels for prime q and binary-modulated channels for q that is a power of 2. For the well-defined q-ary-input symmetric-output channel, we prove that under the Gaussian assumption, the density distribution for messages undergoing decoding is fully characterized by (q-1) quantities. Assuming uniform edge weights, we further show that the density of messages computed by the check node decoder (CND) is fully defined by a single number. We then present the approximate density evolution for regular and irregular LDPC codes, and show that the (q-1) -dimensional integration involved can be simplified using a dimensionality reduction algorithm for the important case of q=2p. Through application of approximate density evolution and linear programming, we optimize the degree distribution of LDPC codes over GF(3) and GF(4). The optimized irregular LDPC codes demonstrate performance close to the Shannon capacity for long codewords. We also design GF(q) codes for high-order modulation by using the idea of a channel adapter. We find that codes designed in this fashion outperform those optimized specifically for the binary additive white Gaussian noise (AWGN) channel for a short codewords and a spectral efficiency of 2 bits per channel use (b/cu).
In this paper, we provide an efficient way to predict iterative belief propagation (BP) decoding thresholds of randomly punctured low-density parity-check (LDPC) code ensembles on the binary-input additive white Gaussian noise channel (AWGNC), given only the BP threshold of the mother code ensemble on the binary erasure channel (BEC) and the code design rate. We show that the predictions are accurate by comparing them with values calculated by discretized density evolution for a variety of puncturing fractions. We find that the strength and suitability of an LDPC code ensemble for random puncturing over the AWGNC with respect to iterative decoding threshold is completely determined by a single constant θ, and this behavior is demonstrated using both LDPC block code and spatially coupled LDPC code ensembles. Finally, we present simulation results that confirm the excellent decoding performance promised by the asymptotic results.