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IEEE International Symposium on Information Theory (ISIT)

Efficient Message-Passing Decoding of LDPC Codes using Vector-based Messages

Mats Tjäder, Mikael Grimnell, Danyo Danev, Hugo Tullberg

Published Jul 1, 2006• 1 citation
Publisher

Abstract

In this paper we investigate reduced complexity decoders for LDPC codes over M-PSK modulation. Capacity calculations show only minor losses if only phase information is considered in the decoder. We device a decoder based on phase angle summation, which unfortunately does not work for codes over Galois fields (but shows promising results for codes over integer rings). For codes over Galois fields we construct table-based decoders which gives performance close to belief propagation decoders at substantially reduced complexity

DOI
10.1109/isit.2006.261647

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In this paper we investigate reduced complexity decoders for LDPC codes over M-PSK modulation. Capacity calculations show only minor losses if only phase information is considered in the decoder. We device a decoder based on phase angle summation, which unfortunately does not work for codes over Galois fields (but shows promising results for codes over integer rings). For codes over Galois fields we construct table-based decoders which gives performance close to belief propagation decoders at substantially reduced complexity
The family of Low Density Parity Check (LDPC) codes is a strong candidate to be used as Forward Error Correction (FEC) in future communication systems due to its strong error correction capability. Most LDPC decoders use the Message Passing algorithm for decoding, which is an iterative algorithm that passes messages between its variable nodes and check nodes. It is not until recently that computation power has become strong enough to make Message Passing on LDPC codes feasible. Although locally simple, the LDPC codes are usually large, which increases the required computation power. Earlier work on LDPC codes has been concentrated on the binary Galois Field, GF(2), but it has been shown that codes from higher order fields have better error correction capability. However, the most efficient LDPC decoder, the Belief Propagation Decoder, has a squared complexity increase when moving to higher order Galois Fields. Transmission over a channel with M-PSK signalling is a common technique to increase spectral efficiency. The information is transmitted as the phase angle of the signal. The focus in this Master’s Thesis is on simplifying the Message Passing decoding when having inputs from M-PSK signals transmitted over an AWGN channel. Symbols from higher order Galois Fields were mapped to M-PSK signals, since M-PSK is very bandwidth efficient and the information can be found in the angle of the signal. Several simplifications of the Belief Propagation has been developed and tested. The most promising is the Table Vector Decoder, which is a Message Passing Decoder that uses a table lookup technique for check node operations and vector summation as variable node operations. The table lookup is used to approximate the check node operation in a Belief Propagation decoder. Vector summation is used as an equivalent operation to the variable node operation. Monte Carlo simulations have shown that the Table Vector Decoder can achieve a performance close to the Belief Propagation. The capability of the Table Vector Decoder depends on the number of reconstruction points and the placement of them. The main advantage of the Table Vector Decoder is that its complexity is unaffected by the Galois Field used. Instead, there will be a memory space requirement which depends on the desired number of reconstruction points.
This study focuses on the efficiency of message-passing-based decoding algorithms for polar and low-density parity-check (LDPC) codes. Both successive cancellation (SC) and belief propagation (BP) decoding algorithms are studied under the message-passing framework. Counter-intuitively, SC decoding demonstrates the highest decoding efficiency although it was considered a weak decoder regarding error-correction performance. We analyze the complexity-performance tradeoff to dynamically track the decoding efficiency, where the complexity is measured by the number of messages passed (NMP), and the performance is measured by the statistical distance to the maximum a posteriori (MAP) estimate. This study offers new insight into the contribution of each message passing in decoding, and compares various decoding algorithms on a message-by-message level.
Decoding Algorithms for Nonbinary LDPC Codes Over GF(q)(q)(q)
In this letter, we address the problem of decoding nonbinary low-density parity-check (LDPC) codes over finite fields GF(q), with reasonable complexity and good performance. In the first part of the letter, we recall the original belief propagation (BP) decoding algorithm and its Fourier domain implementation. We show that the use of tensor notations for the messages is very convenient for the algorithm description and understanding. In the second part of the letter, we introduce a simplified decoder which is inspired by the min-sum decoder for binary LDPC codes. We called this decoder extended min-sum (EMS). We show that it is possible to greatly reduce the computational complexity of the check-node processing by computing approximate reliability measures with a limited number of values in a message. By choosing appropriate correction factors or offsets, we show that the EMS decoder performance is quite good, and in some cases better than the regular BP decoder. The optimal values of the factor and offset correction are obtained asymptotically with simulated density evolution. Our simulations on ultra-sparse codes over very-high-order fields show that nonbinary LDPC codes are promising for applications which require low frame-error rates for small or moderate codeword lengths. The EMS decoder is a good candidate for practical hardware implementations of such codes
In this paper, we propose a binary message-passing algorithm for decoding low-density parity-check (LDPC) codes. The algorithm substantially improves the performance of purely hard-decision iterative algorithms with a small increase in the memory requirements and the computational complexity. We associate a reliability value to each nonzero element of the code's parity-check matrix, and differentially modify this value in each iteration based on the sum of the extrinsic binary messages from the check nodes. For the tested random and finite-geometry LDPC codes, the proposed algorithm can achieve performance as close as 1.3 dB and 0.7 dB to that of belief propagation (BP) at the error rates of interest, respectively. This is while, unlike BP, the algorithm does not require the estimation of channel signal to noise ratio. Low memory and computational requirements and binary message-passing make the proposed algorithm attractive for high-speed low-power applications.